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12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Forced oscillations" }}{PARA 0 " " 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 " " 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 324 7 "DEsol.m" }{TEXT -1 32 " \+ is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It ca n be read into a Maple session by a command similar to the one that fo llows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Applying an external force to a damped system" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 455 116 116 {PLOTDATA 2 "6+-%'CURVESG6%7D7$\"\"!$\"\"&!\"\"7$$F*!\"#F)7$$ \"\"\"F+$\"\")F+7$$\"\"#F+F57$$\"\"$F+F27$$\"\"%F+F57$F)F27$$\"\"'F+F5 7$$\"\"(F+F27$F2F57$$\"\"*F+F27$F1F57$$\"#6F+F27$$\"#7F+F57$$\"#8F+F27 $$\"#9F+F57$$\"#:F+F27$$\"#;F+F57$$\"#F+F27$F6F 57$$\"#@F+F27$$\"#AF+F57$$\"#BF+F27$$\"#CF+F57$$\"#DF+F27$$\"#EF+F57$$ \"#FF+F27$$\"#GF+F57$$\"#HF+F27$F9F57$$\"$0$F.F)7$$\"#JF+F)-%*THICKNES SG6#F6-%&COLORG6&%$RGBGF(F?F(-F$6%7$7$$F.F+F(7$F*F(Faq-Feq6&FgqF5F5F5- F$6'7$7$$!\"$F+FJ7$F5FJ7%7$F0$\"$:\"F.Ffr7$F0$\"$0\"F.7%7$F\\rFirFcr7$ F\\rF\\sFaq-Feq6&FgqF0F0FB-%)POLYGONSG6$7'7$F($F1F.7$F($\"#**F.7$$F+F+ Fjs7$F]tFhsFgs-Feq6&FgqF?F?F?-Fds6$7'7$F_q$F6F.7$F_q$\"#)*F.7$$\"#OF+F gt7$FjtFetFdt-Feq6&FgqF2F2F2-Fds6$7'7$$\"#XF+F(7$FcuF17$F*F1F]rFbu-Feq 6&FgqF2F)F(-F$6&7%7$F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "Transient and steady-state parts of a solution" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 149 "A weight attached to a spring exper iences a resistance to motion and also an external force so that the d ifferential equation governing its motion is" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*x/(d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$% \"tGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+2*x = sin*t +2*cos*t;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"#F'%\"xGF'F',&*&%$sinG F'%\"tGF'F'*(F+F'%$cosGF'F0F'F'" }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 "x(0) = 4" "6#/-%\"xG6#\"\"!\"\"% " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 2;" "6#/-%%e valG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "To solve \+ this initial value problem first consider the homogeneous differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/( d*t^2)+2;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(F&F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+2*x = 0;" "6#/,&*&%#dxG\"\"\"%#dt G!\"\"F'*&\"\"#F'%\"xGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*m+2 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&F'F(F& F(F(F'F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Complet ing the square gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*m+1 = -2+1;" "6#/,(*$%\"mG\"\"#\"\"\"*&F'F(F&F(F(F(F(,&F'! \"\"F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+1)^2 = -1;" "6#/*$ ,&%\"mG\"\"\"F'F'\"\"#,$F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m+1 = ``;" "6#/,&%\"mG\"\"\"F&F&%!G" }{TEXT 310 1 "+" }{TEXT -1 1 " \+ " }{TEXT 313 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "Thi s gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m = -1; " "6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 311 1 "+" }{TEXT -1 2 " " }{TEXT 314 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The complementary solution is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c(t) = C[1]*exp(-t)*c os*t+C[2]*exp(-t)*sin*t;" "6#/-%\"cG6#%\"tG,&**&%\"CG6#\"\"\"F--%$expG 6#,$F'!\"\"F-%$cosGF-F'F-F-**&F+6#\"\"#F--F/6#,$F'F2F-%$sinGF-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The appropriate form for a trial particular solution is: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\" xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*t+B*sin*t;" "6#/%!G,&*(%\" AG\"\"\"%$cosGF(%\"tGF(F(*(%\"BGF(%$sinGF(F*F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$p~'G6 #%\"tG" }{XPPEDIT 18 0 "`` = -A*sin*t+B*cos*t;" "6#/%!G,&*(%\"AG\"\"\" %$sinGF(%\"tGF(!\"\"*(%\"BGF(%$cosGF(F*F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2) = `p ''`(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\" *&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -A*cos*t-B*si n*t;" "6#/%!G,&*(%\"AG\"\"\"%$cosGF(%\"tGF(!\"\"*(%\"BGF(%$sinGF(F*F(F +" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tG F%F'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#d tG!\"\"" }{TEXT -1 5 " and " }{TEXT 309 1 "x" }{TEXT -1 44 " in the se cond differential equation gives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(-A*cos*t-B*sin*t)+ 2*( -A*sin*t+B*cos*t)+2*( A*cos*t+B*sin*t)=sin*t+2*cos*t" "6#/,(-%!G6# ,&*(%\"AG\"\"\"%$cosGF+%\"tGF+!\"\"*(%\"BGF+%$sinGF+F-F+F.F+*&\"\"#F+, &*(F*F+F1F+F-F+F.*(F0F+F,F+F-F+F+F+F+*&F3F+,&*(F*F+F,F+F-F+F+*(F0F+F1F +F-F+F+F+F+,&*&F1F+F-F+F+*(F3F+F,F+F-F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "By comparing the \+ coefficients of the cosine terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-A+2*B+2*A=2" "6#/,(%\"AG! \"\"*&\"\"#\"\"\"%\"BGF)F)*&F(F)F%F)F)F(" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2*B+A = 2;" "6#/,&*&\"\"#\"\"\"%\"BGF'F'%\"AGF'F&" } {TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "By comparing the coefficients of the sine terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "-B-2*A+2*B = 1;" "6#/,(%\"BG!\"\"*&\"\"#\"\"\"%\"AGF)F& *&F(F)F%F)F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B-2*A = 1;" "6# /,&%\"BG\"\"\"*&\"\"#F&%\"AGF&!\"\"F&" }{TEXT -1 16 " ------- (ii). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Multi plying equation (ii) by 2 and subtracting from equation (i) gives " } {XPPEDIT 18 0 "5*A=0" "6#/*&\"\"&\"\"\"%\"AGF&\"\"!" }{TEXT -1 10 ", s o that " }{XPPEDIT 18 0 "A=0" "6#/%\"AG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 21 "It then follows that " }{XPPEDIT 18 0 "B= 1" "6#/%\"BG\"\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular solutio n " }{XPPEDIT 18 0 "p(t) = sin*t;" "6#/-%\"pG6#%\"tG*&%$sinG\"\"\"F'F* " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 " general solution" }{TEXT -1 47 " for the second differential equation \+ given by " }{XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6# F'\"\"\"-%\"pG6#F'F," }{TEXT -1 14 " is therefore " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = C[1]*exp(-t)*cos*t+C[2]*exp(-t )*sin*t+sin*t" "6#/-%\"xG6#%\"tG,(**&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\" F-%$cosGF-F'F-F-**&F+6#\"\"#F--F/6#,$F'F2F-%$sinGF-F'F-F-*&F;F-F'F-F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "x(0) = 4;" "6#/-%\"xG6#\"\"!\"\"%" }{TEXT -1 18 ", it follows that \+ " }{XPPEDIT 18 0 "C[1] = 4;" "6#/&%\"CG6#\"\"\"\"\"%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 4*exp(-t)*cos*t+C[2]*exp(-t)*sin*t+sin*t " "6#/-%\"xG6#%\"tG,(**\"\"%\"\"\"-%$expG6#,$F'!\"\"F+%$cosGF+F'F+F+** &%\"CG6#\"\"#F+-F-6#,$F'F0F+%$sinGF+F'F+F+*&F:F+F'F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=-4*exp(-t)*cos*t-4*exp(-t)*sin*t-C[2]*e xp(-t)*sin*t+C[2]*exp(-t)*cos*t+cos*t" "6#/*&%#dxG\"\"\"%#dtG!\"\",,** \"\"%F&-%$expG6#,$%\"tGF(F&%$cosGF&F0F&F(**F+F&-F-6#,$F0F(F&%$sinGF&F0 F&F(**&%\"CG6#\"\"#F&-F-6#,$F0F(F&F6F&F0F&F(**&F96#F;F&-F-6#,$F0F(F&F1 F&F0F&F&*&F1F&F0F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Eval(dx/dt,t = 0) = 2;" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\"#" } {TEXT -1 17 ", it follws that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2=-4+C[2]+1" "6#/\"\"#,(\"\"%!\"\"&%\"CG6#F$\"\"\"F+F+ " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C[2] = 5;" "6#/&%\"CG6#\"\" #\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "Hence the solu tion to the initial value problem is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 4*exp(-t)*cos*t+5*exp(-t)*sin*t+sin*t;" " 6#/-%\"xG6#%\"tG,(**\"\"%\"\"\"-%$expG6#,$F'!\"\"F+%$cosGF+F'F+F+**\" \"&F+-F-6#,$F'F0F+%$sinGF+F'F+F+*&F7F+F'F+F+" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 312 22 "______________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The complementary solution " }{XPPEDIT 18 0 "c(t) = 4*exp (-t)*cos*t+5*exp(-t)*sin*t;" "6#/-%\"cG6#%\"tG,&**\"\"%\"\"\"-%$expG6# ,$F'!\"\"F+%$cosGF+F'F+F+**\"\"&F+-F-6#,$F'F0F+%$sinGF+F'F+F+" }{TEXT -1 98 " (with the values of the constants determined by the initial co nditions substituted) has a factor " }{XPPEDIT 18 0 "exp(-t);" "6#-%$e xpG6#,$%\"tG!\"\"" }{TEXT -1 22 ", which means that it " }{TEXT 259 13 "tends to zero" }{TEXT -1 4 " as " }{TEXT 307 1 "t" }{TEXT -1 20 " \+ tends to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 33 "For this reason it is called the " }{TEXT 259 14 "transient part" }{TEXT -1 18 " of the \+ solution " }{XPPEDIT 18 0 "phi[trans](t)=4*exp(-t)*cos*t+5*exp(-t)*si n*t" "6#/-&%$phiG6#%&transG6#%\"tG,&**\"\"%\"\"\"-%$expG6#,$F*!\"\"F.% $cosGF.F*F.F.**\"\"&F.-F06#,$F*F3F.%$sinGF.F*F.F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {TEXT 308 1 "t" }{TEXT -1 26 " is \"large\" the affect of " }{XPPEDIT 18 0 "c(t);" "6#-%\"cG6#%\"tG" }{TEXT -1 107 " becomes neglible, and t he behaviour of the sytem is then essentially described by the particu lar solution " }{XPPEDIT 18 0 "p(t) = sin*t;" "6#/-%\"pG6#%\"tG*&%$sin G\"\"\"F'F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "This is ca lled the " }{TEXT 259 12 "steady-state" }{TEXT -1 23 " part of the sol ution " }{XPPEDIT 18 0 "phi[sstate](t) = sin*t;" "6#/-&%$phiG6#%'ssta teG6#%\"tG*&%$sinG\"\"\"F*F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The graph of the solution is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 106 "g := t -> 4*exp(-t)*cos(t)+5*exp(-t)*sin(t)+s in(t):\n'g(t)'=g(t);\nplot(g(t),t=0..8*Pi,labels=[`t`,`x(t)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,(*(\"\"%\"\"\"-%$expG6# ,$F'!\"\"F+-%$cosGF&F+F+*(\"\"&F+F,F+-%$sinGF&F+F+F5F+" }}{PARA 13 "" 1 "" {GLPLOT2D 450 198 198 {PLOTDATA 2 "6%-%'CURVESG6$7[u7$$\"\"!F)$\" \"%F)7$$\"3'HqLi#zxZo!#>$\"3b/ZpPYS9T!#<7$$\"3gSnC&eb&p8!#=$\"3#*Q$)*y +qr=%F27$$\"3gD%e:[W>r\"F6$\"3Uuh6!*=\\4UF27$$\"3!46qyPLV0#F6$\"3'[Nd( eWABUF27$$\"3='z\"=uAs'R#F6$\"3-20II0c\"F2$ \"3i%z;#>'R'e?F27$$\"3*>YD>QK.$=F2$\"3eT'y%>I%od\"F27$$\"3LLPMjW8+@F2$ \"3(33%eIjCW6F27$$\"3k$>lDAa'oBF2$\"3s%\\u/:V+d(F67$$\"3(Rl'y\")R6_b\"4kE)F67$$\"3+04AQK\"Q=%F2$!3^6N8p 0r+'*F67$$\"3e!=J%*o\"o;VF2$!3?l(\\UBj[+\"F27$$\"3;c9kS,b\\WF2$!33$3_P &>@M5F27$$\"3]$fYiO%)f^%F2$!3)3BS`q@H/\"F27$$\"3%3t^=f=Ce%F2$!33!pM]U) eZ5F27$$\"3=ooX&[F2$!36C0HxOhC5F27$$\"3IP)=/6I'))\\F2$!39pYr@hNu)*F67 $$\"3J_s4W;ID^F2$!3B&[qM$eQO$*F67$$\"3KncxxJ(>E&F2$!3_N0w<2;S')F67$$\" 3O'H(G+Rt-bF2$!3g!Qh,9QN1(F67$$\"3JE*)zAY\\VdF2$!34(p#*HD)36^F67$$\"3b ')\\KsT`9gF2$!3)eh\\A)*>Df#F67$$\"3!o/^=stbG'F2$\"3o\"*3/n.fi)*!#?7$$ \"3%RS3e=Exb'F2$\"3!G]TnFD\\y#F67$$\"3)>wl(\\'y)HoF2$\"3G(>>8Z*oj_F67$ $\"3kSRTxz9#4(F2$\"3sft+@Ew)G(F67$$\"3W=@10tTatF2$\"3r=b/17D=))F67$$\" 3a+(RF9+NZ(F2$\"37;W2Q>m>$*F67$$\"3u\"G1P')F2$\"3EBI![(*=()3(F67$$\"3C#[1tArp())F2$\"3(f&e .\"GIq?&F67$$\"3Q)G#HKF/c\"*F2$\"353d_gFX_EF67$$\"3t#4ysB9^V*F2$!3'==) y\"=of1\"F/7$$\"3Ios.1)R@o*F2$!3wxgMV!4&[DF67$$\"3!RW'zu`;H**F2$!36%e& o-kfN[F67$$\"3pur^d\">+-\"!#;$!3HLf*HTxC+(F67$$\"3+0Zbx<7Z5Fi^l$!3IiQz5Fi^l$!3?*3_?hL$)z*F67$$\"3e%H&yKR$e3\"Fi ^l$!3?m!ox&*Qq!**F67$$\"3cfqlefG#4\"Fi^l$!35[yn0m_u**F67$$\"3OC)GX)zt) 4\"Fi^l$!3$oqpja^++\"F27$$\"3@]Sm'**pa5\"Fi^l$!3u=XYPxF$)**F67$$\"3*eF *z3??76Fi^l$!3g.EITC#3#**F67$$\"3u,X$4-M*=6Fi^l$!3voL`4@V8)*F67$$\"3fF (pI.mc7\"Fi^l$!3=N/!)=Gfh'*F67$$\"3Iz,Md+8R6Fi^l$!3*os:_H:vA*F67$$\"3# 3j5;3%f_6Fi^l$!3>^T&=@Vki)F67$$\"3gq\")>OCKx6Fi^l$!3#)*=sMe+d7(F67$$\" 3N5dy!z]??\"Fi^l$!3c([Fon1:>&F67$$\"3KRI*=mB(G7Fi^l$!3y*z_l!*=^v#F67$$ \"37o.+LlRb7Fi^l$!3m%)pixi5R7F/7$$\"3iCq!)\\@5$G\"Fi^l$\"3*z%*Q9cheh#F 67$$\"3I\"o8mw23J\"Fi^l$\"3)pfETy:h:&F67$$\"3]M&*[(eD\\L\"Fi^l$\"3I]** )>,yL0(F67$$\"3_(Ql$3M/f8Fi^l$\"3]yvXkhNU&)F67$$\"31&*H+it1s8Fi^l$\"3? dXn\\,?X\"*F67$$\"3i-1k:84&Q\"Fi^l$\"3C#p;S2TJf*F67$$\"3I1%fCH.;R\"Fi^ l$\"3o!)3L5bac(*F67$$\"3+5#y#p_6)R\"Fi^l$\"3%RLVza\"fy)*F67$$\"3o8q4Ys i/9Fi^l$\"3-yBE:=we**F67$$\"3b)3?XH*F67$$\"3SLb#pHf\\Y\"Fi^l$\"3 yuUc&\\'e:()F67$$\"31t^:=bG\"\\\"Fi^l$\"3+0:O'[&RRrF67$$\"3*G\"[QR\"*=-(G^*o\"Fi^l$!3A>n5n-7t#*F67$$\" 34E!z=rzIq\"Fi^l$!3GlM=q$**F67$$\"3u(>pV(*HMs\"Fi^l$!3as3NI*>,** *F67$$\"3>b#*>&R8-t\"Fi^l$!3i0`mA)ps***F67$$\"3@geo#=gjt\"Fi^l$!3I\"QX &)*=.k**F67$$\"3BlC[J$=Fi^l$!3'3zN&[x2_\\F67$$\"3[%H)=8/ We=Fi^l$!3H\"f>aMd0i#F67$$\"3_Xp'GjKP)=Fi^l$!3ZV/v0i#HA\"F/7$$\"3%p(H0 !Hq,\">Fi^l$\"3mcb!)zk$[\\#F67$$\"3O3!Rs%zgO>Fi^l$\"3q*o3@N-'Q\\F67$$ \"3CI@]*ebC'>Fi^l$\"3E*HEZvgr*pF67$$\"37__wJKI))>Fi^l$\"3C`!f9MN3f)F67 $$\"3T\"fsZFL=+#Fi^l$\"3c[n=#)ps-#*F67$$\"3MI*zxJj`,#Fi^l$\"3!>^Y'\\%3 kk*F67$$\"3***f$GR$G@-#Fi^l$\"3\"Q,(*>_6D!)*F67$$\"3GpsygL*)G?Fi^l$\"3 (o*z>-+x8**F67$$\"3#*Q4H#Qec.#Fi^l$\"3-%41#Fi^l$\"3'4W'Q)Gg>!)*F67$$\"3' eD%z#[&[o?Fi^l$\"3%\\#**>WmA_'*F67$$\"3;zSHAl^\"3#Fi^l$\"3F$=&H*>&oI#* F67$$\"3!G!Rzhva%4#Fi^l$\"3mJ#fW-@El)F67$$\"3b<0!=!)*>@@Fi^l$\"3E^([^& GvEqF67$$\"3%>82=/_y9#Fi^l$\"3D!)RBuEp/\\F67$$\"3Qm+\"ye$Gu@Fi^l$\"3'G uv3e(odCF67$$\"3V+I\"Q8:2?#Fi^l$!33Yf'>63-g\"F/7$$\"3y@b^zH+DAFi^l$!3A d(>f9#**fDF67$$\"37V!=_#3H\\AFi^l$!38#e#[%e!p4[F67$$\"3-9^FPu7xAFi^l$! 3$38))oX)oLqF67$$\"3$\\=K$\\S'\\I#Fi^l$!36Q[X@B<;()F67$$\"33Qa;[FT$e95BM#Fi^l$!3%G_3wBnP!**F67$$\"31q&[_\\ M&[BFi^l$!3ayzL%ew1(**F67$$\"3Y'>lY%)eZN#Fi^l$!3#QYh;[p*)***F67$$\"3)z cp$*H&RhBFi^l$!3%>FPF%)yk)**F67$$\"3_RR2a<.oBFi^l$!3wqIrj8-I**F67$$\"3 /6$y(3#oYP#Fi^l$!3zcsO1c%)H)*F67$$\"3e#o#[jYI\"Q#Fi^l$!3!*\\H=3ER'o*F6 7$$\"3kD9*GdxXR#Fi^l$!31He`G*oBF*F67$$\"30p,I#[]yS#Fi^l$!37g%[L[L_p)F6 7$$\"3qi'y!=#eKV#Fi^l$!3Cnm3;*oY<(F67$$\"3Mcr&Q&fmeCFi^l$!3?$HJypFM>&F 67$$\"3EyN&e.qf[#Fi^l$!3g*)Q'4,4mp#F67$$\"3<++&y6uK^#Fi^l$!3(4r1:!)pp, &!#D-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"tG%%x(t)G-%%VIE WG6$;F($\"+BTF8D!\")%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "We can plot the transient part of the sol ution " }{XPPEDIT 18 0 "phi[trans](t)=4*exp(-t)*cos*t+5*exp(-t)*sin*t " "6#/-&%$phiG6#%&transG6#%\"tG,&**\"\"%\"\"\"-%$expG6#,$F*!\"\"F.%$co sGF.F*F.F.**\"\"&F.-F06#,$F*F3F.%$sinGF.F*F.F." }{TEXT -1 40 ", the st eady-state part of the solution " }{XPPEDIT 18 0 "phi[sstate](t) = sin *t;" "6#/-&%$phiG6#%'sstateG6#%\"tG*&%$sinG\"\"\"F*F-" }{TEXT -1 45 " \+ in the same picture along with the solution " }{XPPEDIT 18 0 "g(t)=phi [trans](t)+phi[sstate](t)" "6#/-%\"gG6#%\"tG,&-&%$phiG6#%&transG6#F'\" \"\"-&F+6#%'sstateG6#F'F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 374 "phi[trans] := t -> 4 *exp(-t)*cos(t)+5*exp(-t)*sin(t):\nphi[sstate] := t -> sin(t):\ng := u napply(phi[trans](t)+phi[sstate](t),t):\n'phi[trans](t)'=phi[trans](t) ;\n'phi[sstate](t)'=phi[sstate](t);\n'g(t)'=g(t);\nplot([g(t),phi[tran s](t),phi[sstate](t)],t=0..4*Pi,color=[red,blue,magenta],\n legend=[`c omplete solution`,`transient part of solution`,`steady-state part of s olution`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$phiG6#%&transG6#% \"tG,&*(\"\"%\"\"\"-%$expG6#,$F*!\"\"F.-%$cosGF)F.F.*(\"\"&F.F/F.-%$si nGF)F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$phiG6#%'sstateG6#%\"t G-%$sinGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,(*(\"\"% \"\"\"-%$expG6#,$F'!\"\"F+-%$cosGF&F+F+*(\"\"&F+F,F+-%$sinGF&F+F+F5F+ " }}{PARA 13 "" 1 "" {GLPLOT2D 657 390 390 {PLOTDATA 2 "6'-%'CURVESG6% 7go7$$\"\"!F)$\"\"%F)7$$\"3qck?EzxZo!#>$\"3=MVpPYS9T!#<7$$\"3M\"HT_eb& p8!#=$\"3[8z*y+qr=%F27$$\"3Q9;b\"[W>r\"F6$\"3E=e6!*=\\4UF27$$\"3;P>'yP LV0#F6$\"3#[7d(eWABUF27$$\"3#*fA-PIF6$\"3ir3L!o+*>UF27$$\"3?Y[NmE$\\ L$F6$\"3SI'p.x2x?%F27$$\"3ox4H9M%Gj$F6$\"3u')*4SZ!)3>%F27$$\"3;4rAiTvI RF6$\"3QJ#y_VA(pTF27$$\"3os$*4ecdEXF6$\"3`9QQp)pb6%F27$$\"3>O;(R:(RA^F 6$\"37vSN6kWZSF27$$\"3ss#Q#GV_ikF6$\"3G$z5Z[HL&QF27$$\"383\\]-:l-yF6$ \"3Gv.os1F=OF27$$\"3)z*3f4>m^\"*F6$\"3kd*yrBFjN$F27$$\"3o)on;Bn+0\"F2$ \"3wxAcR4[\"3$F27$$\"3e%)yF6rK%=\"F2$\"3<-)zbHdW!GF27$$\"3^!3))3*pe=8F 2$\"3G-[Z(yP1`#F27$$\"3W4wu&3jIW\"F2$\"3I0!4$[Mg$G#F27$$\"3QQrg!=Rvc\" F2$\"3Q!H\\RC&\\X?F27$$\"3rBw=`ZJD=F2$\"3UX:\")[\\M&e\"F27$$\"3%*G@5>m !>4#F2$\"3]b?^Wgvc6F27$$\"3EA;_@NkdBF2$\"32mZRzGJ@xF67$$\"3flt())e')4j #F2$\"3QG\"==-uO8%F67$$\"3uO!)Q6turGF2$\"3cbv&\\p#=Q7F67$$\"3t=I\"4'oy UJF2$!3(>%G\\&y65u\"F67$$\"3k$Hp[KR\\T$F2$!3;'RGUWZ'4WF67$$\"3!yU;Dl3s n$F2$!3Y$*GKw4@>mF67$$\"3)QkqyKu`\"RF2$!3im@'3_i_D)F67$$\"3C>Tu^jd)>%F 2$!3%z7R&*Q)*yl*F67$$\"3O])*o#)4`=VF2$!3PR8HZ\"z`+\"F27$$\"3e!eNOh&[QW F2$!3%*R33r;PK5F27$$\"3kFz7m8-yXF2$!3!>SDv?0u/\"F27$$\"3gv-i=rbK*>&)*>&**F67$$\"3C\"elvy&4+^F2$!3QJ#[ZNzxW*F67$$\"3i$p_x))3cB&F2$! 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35sp5eL9\"R\"F67$Fgx$!3h,*f1V-E?'F[y7$$\"3%of4!\\2^:kF2$\"3$3#RO&4'R>8 F67$F]y$\"3-e)p\"G%Rbn#F67$$\"3996UPziumF2$\"3vm(F2$\"3k[()*3Hlp(yF67$F\\z$\"391w$==(=K')F67$Faz$\"33O,o#Q2,A* F67$Ffz$\"3QrC,TO]['*F67$$\"3O(\\l'G'Q(**F67$$\"3yfX4e%Q4\"yF2$\"3#Q(G>byt!***F67$F`[l$\"3MiH*pWs $****F67$$\"3u\"*4`x78yyF2$\"3w\")yH:T3(***F67$$\"3v6;3JT[8zF2$\"3]\\z o'[-B)**F67$$\"3wJAj%)p$)[zF2$\"3)=0xO.Y]&**F67$Fe[l$\"3CqL\\=)[`\"**F 67$$\"3-!4%GXb*[0)F2$\"3-M&eU-Y))z*F67$Fj[l$\"3b7'\\MHwLj*F67$F_\\l$\" 3WYb!oO]F67$$\"3Z4%o9sR8.*F2$ \"3*z@EM?iO$QF67$Fc]l$\"3A*p`w#z\\hDF67$$\"3/,X!f1-AH*F2$\"3C/84*oy=K \"F67$Fh]l$\"3A@e2m$eZ6'F[y7$$\"33$)oA]9&3b*F2$!3[[))z]xRd7F67$F]^l$!3 ;Wv<:O+aDF67$$\"3U5;ZZzF7)*F2$!3;#G@[W\\(yPF67$Fb^l$!3`A?.`%o/%\\F67$$ \"3wbf))e;o25Fi^l$!3[,[Fe13ogF67$Fg^l$!3;#fVJoyZ3(F67$$\"3#G,$RTFCM5Fi ^l$!3rz\"o\"[TvTzF67$F]_l$!3YiB11I1k')F67$Fb_l$!3Aea&e]/2D*F67$Fg_l$!3 S4>;+*4Ln*F67$$\"3E++S2R`!3\"Fi^l$!3d\"\\*ydyf>)*F67$F\\`l$!3];MQh\\-B **F67$$\"3'41^rtX/4\"Fi^l$!3g$)*Hk#***F67$$\"3@GxKG&Hk5\"Fi^l$!3'y2 x]T'Rw**F67$$\"3IJ0/4bY46Fi^l$!3OW#p]6b4&**F67$Ff`l$!3H5n>YDM;**F67$$ \"3cS*y6Xt&=6Fi^l$!3KTQUF'R(>)*F67$F[al$!3;x\")*=rVpo*F67$F`al$!3c`eXY 9&)[#*F67$Feal$!3eYlK!=!)=j)F67$$\"3A5Z\\B2$\\;\"Fi^l$!3&GK.7G\">QzF67 $Fjal$!3:fY\"Fi^l$!3&yejj5i*HhF67$F_bl$!388Su0 \"[/.&F67$$\"3g!\\M!4\"Hm@\"Fi^l$!3=p%[a(e\"\\*QF67$Fdbl$!3+Y[/q(3mp#F 67$$\"3SV=#z,&)HC\"Fi^l$!3%y7T1cn4O\"F67$Fibl$!3BXiH,t\"f^#!#D-F_cl6&F aclFbclF(Fbcl-Ffcl6#%>steady-state~part~of~solutionG-%+AXESLABELSG6$Q \"t6\"Q!Fd`o-%%VIEWG6$;F($\"+iqjc7Fdcl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 42.000000 0 1 "complete solution" "transient pa rt of solution" "steady-state part of solution" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "6 < t;" "6#2\"\"'%\"tG" } {TEXT -1 66 " the solution is indistinguishable from the steady-state \+ solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Solution us ing " }{TEXT 0 6 "dsolve" }{TEXT -1 5 " and " }{TEXT 0 7 "desolve" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "de := diff(x(t),t$2)+2*diff(x(t),t)+2*x(t)=sin( t)+2*cos(t);\nic := x(0)=4,D(x)(0)=2;\ndsolve(\{de,ic\},x(t));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#%\"tG-%\"$G 6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&F1F2F*F2F2,&-%$sinGF,F2*&F1F2-%$co sGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!\"\" %/--%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG ,(-%$sinGF&\"\"\"*(\"\"%F+-%$expG6#,$F'!\"\"F+-%$cosGF&F+F+*(\"\"&F+F. F+F)F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "desolve" }{TEXT -1 24 " can be used in stead of " }{TEXT 0 6 "dsolve" }{TEXT -1 45 " if we want to see the st eps in the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "de := diff(x(t),t$2)+2*diff(x(t),t)+2*x( t)=sin(t)+2*cos(t);\nic := x(0)=4,D(x)(0)=2;\ndesolve(\{de,ic\},info=t rue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"xG6#% \"tG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&F1F2F*F2F2,&-%$sinGF,F2* &F1F2-%$cosGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6 #\"\"!\"\"%/--%\"DG6#F(F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8au xiliary~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&F)F*F(F*F*F)F*\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%+roots~.~.~G^$!\"\"\"\"\"^$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa mple 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 315 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 127 "An oscillatory system is subject to damping and to an external dr iving force. The differential equation governing its motion is" }} {PARA 256 "" 0 "" {TEXT -1 3 "16 " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+8;" " 6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\")F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+65*x = 55*sin*3*t-103*cos*3*t;" "6#/,&*&%# dxG\"\"\"%#dtG!\"\"F'*&\"#lF'%\"xGF'F',&**\"#bF'%$sinGF'\"\"$F'%\"tGF' F'**\"$.\"F'%$cosGF'F1F'F2F'F)" }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 "x(0) = 4" "6#/-%\"xG6#\"\"!\"\"% " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 2;" "6#/-%%e valG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 53 "(a) Find the solution of this initial val ue problem. " }}{PARA 0 "" 0 "" {TEXT -1 67 "(b) Identify the transien t and steady-state parts of the solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 90 "To solve this initial value problem first consider the homogeneous differential equation: " }}{PARA 256 "" 0 " " {TEXT -1 4 " 16 " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+8;" "6#,&*(%\"dG\"\" #%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\")F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt+65*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"#lF '%\"xGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The a uxiliary equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "16*m^2+8*m+65 = 0;" "6#/,(*&\"#;\"\"\"*$%\"mG\"\"#F'F'*&\"\")F'F )F'F'\"#lF'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+m/2+65/16=0" "6 #/,(*$%\"mG\"\"#\"\"\"*&F&F(F'!\"\"F(*&\"#lF(\"#;F*F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Completing the square gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+m/2+1/16 = -65/16 +1/16;" "6#/,(*$%\"mG\"\"#\"\"\"*&F&F(F'!\"\"F(*&F(F(\"#;F*F(,&*&\"#lF (F,F*F**&F(F(F,F*F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "th at is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+1/4)^2 \+ = -4;" "6#/*$,&%\"mG\"\"\"*&F'F'\"\"%!\"\"F'\"\"#,$F)F*" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m+1/4 = ``;" "6#/,&%\"mG\"\"\"*&F&F&\"\"%!\" \"F&%!G" }{TEXT 304 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*i;" "6#*& \"\"#\"\"\"%\"iGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "Th is gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m = -1/ 4;" "6#/%\"mG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 1 " " }{TEXT 305 1 "+ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*i;" "6#*&\"\"#\"\"\"%\"iGF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The complementary solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c(t) = C[1]*exp(-t/4)*cos*2*t+C[2]*exp( -t/4)*sin*2*t;" "6#/-%\"cG6#%\"tG,&*,&%\"CG6#\"\"\"F--%$expG6#,$*&F'F- \"\"%!\"\"F4F-%$cosGF-\"\"#F-F'F-F-*,&F+6#F6F--F/6#,$*&F'F-F3F4F4F-%$s inGF-F6F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 57 "The appropriate form for a trial particul ar solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*3*t+B*sin* 3*t;" "6#/%!G,&**%\"AG\"\"\"%$cosGF(\"\"$F(%\"tGF(F(**%\"BGF(%$sinGF(F *F(F+F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6 #/*&%#dxG\"\"\"%#dtG!\"\"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "`` = -3*A*sin *3*t+3*B*cos*3*t;" "6#/%!G,&*,\"\"$\"\"\"%\"AGF(%$sinGF(F'F(%\"tGF(!\" \"*,F'F(%\"BGF(%$cosGF(F'F(F+F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d ^2*x/(d*t^2) = `p ''`(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F (!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -9*A*cos*3*t-9*B*sin*3*t;" "6 #/%!G,&*,\"\"*\"\"\"%\"AGF(%$cosGF(\"\"$F(%\"tGF(!\"\"*,F'F(%\"BGF(%$s inGF(F+F(F,F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Subst ituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\" \"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dx/dt" "6#* &%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 5 " and " }{TEXT 303 1 "x" }{TEXT -1 37 " in the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "16*( -9*A*cos*3*t-9*B*sin*3*t)+8*(-3*A* sin*3*t+3*B*cos*3*t)+65*(A*cos*3*t+B*sin*3*t)=55*sin*3*t-103*cos*3*t" "6#/,(*&\"#;\"\"\",&*,\"\"*F'%\"AGF'%$cosGF'\"\"$F'%\"tGF'!\"\"*,F*F'% \"BGF'%$sinGF'F-F'F.F'F/F'F'*&\"\")F',&*,F-F'F+F'F2F'F-F'F.F'F/*,F-F'F 1F'F,F'F-F'F.F'F'F'F'*&\"#lF',&**F+F'F,F'F-F'F.F'F'**F1F'F2F'F-F'F.F'F 'F'F',&**\"#bF'F2F'F-F'F.F'F'**\"$.\"F'F,F'F-F'F.F'F/" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "By co mparing the coefficients of the cosine terms we obtain the equation: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-144*A+24*B+65*A= -103" "6#/,(*&\"$W\"\"\"\"%\"AGF'!\"\"*&\"#CF'%\"BGF'F'*&\"#lF'F(F'F', $\"$.\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "79*A-24*B=103" "6#/, &*&\"#z\"\"\"%\"AGF'F'*&\"#CF'%\"BGF'!\"\"\"$.\"" }{TEXT -1 15 " ---- --- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "By comparing the coefficients of the sine terms we obtain the equa tion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-144*B-24*A +65*B=55" "6#/,(*&\"$W\"\"\"\"%\"BGF'!\"\"*&\"#CF'%\"AGF'F)*&\"#lF'F(F 'F'\"#b" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "24*A+79*B = -55;" "6# /,&*&\"#C\"\"\"%\"AGF'F'*&\"#zF'%\"BGF'F',$\"#b!\"\"" }{TEXT -1 16 " \+ ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Multiplying equation (i) by 24 and equation (ii) by 79 gi ves the two equations:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1896*A-576*B=2472,``],[1896*A+6241*B= -4345,``])" "6# -%*PIECEWISEG6$7$/,&*&\"%'*=\"\"\"%\"AGF+F+*&\"$w&F+%\"BGF+!\"\"\"%sC% !G7$/,&*&F*F+F,F+F+*&\"%TiF+F/F+F+,$\"%XVF0F2" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 64 "Subtracting the second of these equations from the first gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-6817*B=6817" "6#/,$*&\"%F-F@ F-F'F-F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "x(0) = 4;" "6#/-%\"xG6#\"\" !\"\"%" }{TEXT -1 25 ", we obtain the equation " }{XPPEDIT 18 0 "4=C[1 ]+1" "6#/\"\"%,&&%\"CG6#\"\"\"F)F)F)" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "C[1] = 3;" "6#/&%\"CG6#\"\"\"\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x = 3*exp(-t/4)*cos*3*t+C[2]*exp(-t/4)*sin*2*t+cos*3 *t-sin*3*t;" "6#/%\"xG,**,\"\"$\"\"\"-%$expG6#,$*&%\"tGF(\"\"%!\"\"F0F (%$cosGF(F'F(F.F(F(*,&%\"CG6#\"\"#F(-F*6#,$*&F.F(F/F0F0F(%$sinGF(F6F(F .F(F(*(F1F(F'F(F.F(F(*(F;F(F'F(F.F(F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt = -3/4" "6#/*&%#dxG\"\"\"%#dtG!\"\",$*&\"\"$F&\" \"%F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/4)*cos*3*t-9*exp(-t/4) *sin*3*t-C[2]/4;" "6#,(**-%$expG6#,$*&%\"tG\"\"\"\"\"%!\"\"F-F+%$cosGF +\"\"$F+F*F+F+*,\"\"*F+-F&6#,$*&F*F+F,F-F-F+%$sinGF+F/F+F*F+F-*&&%\"CG 6#\"\"#F+F,F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/4)*sin*2*t+2*C [2]*exp(-t/4)*cos*2*t-3*sin*3*t-3*cos*3*t;" "6#,***-%$expG6#,$*&%\"tG \"\"\"\"\"%!\"\"F-F+%$sinGF+\"\"#F+F*F+F+*.F/F+&%\"CG6#F/F+-F&6#,$*&F* F+F,F-F-F+%$cosGF+F/F+F*F+F+**\"\"$F+F.F+F:F+F*F+F-**F:F+F8F+F:F+F*F+F -" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Eval(dx/dt,t=0)=0" "6#/-%%EvalG6$* &%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 17 ", it follws that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0=-3/4+2*C[2]-3" "6 #/\"\"!,(*&\"\"$\"\"\"\"\"%!\"\"F**&\"\"#F(&%\"CG6#F,F(F(F'F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "C[2] =15/8" "6#/&%\"CG6#\"\"#*&\"#:\"\"\"\"\")!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 52 "Hence the solution to the initial value p roblem is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 3*exp(-t/4)*cos*2*t+15/8;" "6#/-%\"xG6#%\"tG,&*,\"\"$\"\"\"-%$expG6#, $*&F'F+\"\"%!\"\"F2F+%$cosGF+\"\"#F+F'F+F+*&\"#:F+\"\")F2F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/4)*sin*2*t+cos*3*t-sin*3*t;" "6#,(**-% $expG6#,$*&%\"tG\"\"\"\"\"%!\"\"F-F+%$sinGF+\"\"#F+F*F+F+*(%$cosGF+\" \"$F+F*F+F+*(F.F+F2F+F*F+F-" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 306 33 "_________________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 14 "transient part" }{TEXT -1 20 " of the solution is " }{XPPEDIT 18 0 "q(t) = 3*exp(-t/4)*cos*2*t+15/8;" "6#/- %\"qG6#%\"tG,&*,\"\"$\"\"\"-%$expG6#,$*&F'F+\"\"%!\"\"F2F+%$cosGF+\"\" #F+F'F+F+*&\"#:F+\"\")F2F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/4)* sin*2*t;" "6#**-%$expG6#,$*&%\"tG\"\"\"\"\"%!\"\"F,F*%$sinGF*\"\"#F*F) F*" }{TEXT -1 13 ", while the " }{TEXT 259 12 "steady-state" }{TEXT -1 10 " part is " }{XPPEDIT 18 0 "p(t) = cos*3*t-sin*3*t;" "6#/-%\"pG 6#%\"tG,&*(%$cosG\"\"\"\"\"$F+F'F+F+*(%$sinGF+F,F+F'F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "We can plot the transient and steady-state parts of the solution i n the same picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 396 "phi[trans] := t -> 3*exp(-t/4)*cos(2*t)+15 /8*exp(-t/4)*sin(2*t):\nphi[sstate] := t -> cos(3*t)-sin(3*t):\ng := u napply(phi[trans](t)+phi[sstate](t),t):\n'phi[trans](t)'=phi[trans](t) ;\n'phi[sstate](t)'=phi[sstate](t);\n'g(t)'=g(t);\nplot([g(t),phi[tran s](t),phi[sstate](t)],t=0..6*Pi,color=[red,blue,magenta],\n legend=[`c omplete solution`,`transient part of solution`,`steady-state part of s olution`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$phiG6#%&transG6#% \"tG,&*&#\"#:\"\")\"\"\"*&-%$expG6#,$*&\"\"%!\"\"F*F0F8F0-%$sinG6#,$*& \"\"#F0F*F0F0F0F0F0*(\"\"$F0F2F0-%$cosGF;F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$phiG6#%'sstateG6#%\"tG,&-%$cosG6#,$*&\"\"$\"\"\"F* F2F2F2-%$sinGF.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG 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8ii-F%\\18!p69F27$$\"3Q\\]My% )Ra([()QTT\"F27$F__o$!3<1.Rsjl49F27$Fd_o$!3')R2Ja&>+Q\"F27 $Fi_o$!3LnLFm+OB8F27$$\"3yC&HXZY5x\"F`em$!3JD=N]AWM7F27$F^`o$!3u&H;lv^ !=6F27$$\"3S&4'ec6+\"y\"F`em$!3!f))Q#3()yn(*F/7$Fc`o$!3*f06ISww8)F/7$$ \"3-mEkQe&4z\"F`em$!3L6TbSh2(H=F`em$\"3_wud%)=9(4*F/7$F\\bo$ \"3qdr))\\Fma5F27$$\"3'=F`em$\"3uK- ZV^$zS\"F27$Feco$\"3-`#4\"fQd$R\"F27$Fjco$\"337fSfgbb7F27$F_do$\"3Ari \"H6+++\"F2-Fddo6&FfdoFgdoF(Fgdo-F[eo6#%>steady-state~part~of~solution G-%+AXESLABELSG6$Q\"t6\"Q!Fghv-%%VIEWG6$;F($\"+#fb\\)=Fido%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "complete solu tion" "transient part of solution" "steady-state part of solution" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Solution by " }{TEXT 0 6 "dsolve" }{TEXT -1 5 " and " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "de := 16*diff(x(t),t$2)+8*diff(x(t),t)+65*x(t)=-103*cos(3*t)+55 *sin(3*t);\nic := x(0)=4,D(x)(0)=0;\ndsolve(\{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"#;\"\"\"-%%diffG6$-%\"xG6#%\"tG -%\"$G6$F0\"\"#F)F)*&\"\")F)-F+6$F-F0F)F)*&\"#lF)F-F)F),&*&\"$.\"F)-%$ cosG6#,$*&\"\"$F)F0F)F)F)!\"\"*&\"#bF)-%$sinGF@F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!\"\"%/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,**&#\"#:\"\")\"\"\"*&-%$expG6 #,$*&\"\"%!\"\"F'F-F5F--%$sinG6#,$*&\"\"#F-F'F-F-F-F-F-*(\"\"$F-F/F--% $cosGF8F-F--F76#,$*&F=F-F'F-F-F5-F?FAF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "desolv e" }{TEXT -1 24 " can be used instead of " }{TEXT 0 6 "dsolve" }{TEXT -1 45 " if we want to see the steps in the solution." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "de := 16* diff(x(t),t$2)+8*diff(x(t),t)+65*x(t)=-103*cos(3*t)+55*sin(3*t);\nic : = x(0)=4,D(x)(0)=0;\ndesolve(\{de,ic\},info=true);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/,(*&\"#;\"\"\"-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F 0\"\"#F)F)*&\"\")F)-F+6$F-F0F)F)*&\"#lF)F-F)F),&*&\"$.\"F)-%$cosG6#,$* &\"\"$F)F0F)F)F)!\"\"*&\"#bF)-%$sinGF@F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!\"\"%/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*&\"#;\"\"\")% \"mG\"\"#F(F(*&\"\")F(F*F(F(\"#lF(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G^$#!\"\"\"\" %\"\"#^$F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F. -FC6#,$*&\"\"&F.F-F.F.F.F.F,*&#\"#bFF.-F0FIF.F.F.*&FNF.*&F>F.-F 0FDF.F.F.F-F.F.*(F&F.-FCF1F.-F66$,**&#F;FF/F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Afrom~the~initial~conditions~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/\"\"!,(\"\"$!\"\"*&\"\"#\"\"\"&%\"CG6#F*F*F**&#F*\"\"%F*&F,6#F)F*F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%,&\"\"\"F&&%\"CG6#\"\"#F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%-so~that~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"# \"#:\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"xG6#%\"tG,*-%$sinG6#,$*&\"\"$\"\"\"F'F/F/!\"\"-%$cosGF+F/*&#\"#: \"\")F/*&-%$expG6#,$*&\"\"%F0F'F/F0F/-F*6#,$*&\"\"#F/F'F/F/F/F/F/*(F.F /F8F/-F2F?F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 317 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 117 "A weight attached \+ to a spring experiences an external force so that the differential equ ation governing its motion is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+4*x = 5*exp(-t/2)*sin*t;" "6#/,&*(%\"dG\" \"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"\"%F)F(F)F)**\"\"&F)-%$expG6 #,$*&F,F)F'F-F-F)%$sinGF)F,F)" }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%eval G6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "(a) Find the solut ion of this initial value problem. " }}{PARA 0 "" 0 "" {TEXT -1 67 "(b ) Identify the transient and steady-state parts of the solution. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 110 ": In this example there is no damping force, and the for cing function is transitory, that is, it dies away as " }{TEXT 319 1 " t" }{TEXT -1 11 " increases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 318 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 66 "The auxiliary equation for the homogeneous differential e quation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d *t^2)+4*x = 0" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*& \"\"%F)F(F)F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+4=0" "6#/,&*$ %\"mG\"\"#\"\"\"\"\"%F(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "which has the solutions " }{XPPEDIT 18 0 "m=``" "6#/%\"mG %!G" }{TEXT 320 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*i" "6#*&\"\"# \"\"\"%\"iGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 259 22 "complementary solution" }{TEXT -1 38 " for the non-homog eneous equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c(t)=C[1]*cos*2*t+C[2]*sin*2*t" "6#/-%\"cG6#%\"tG,&**&%\"CG6#\"\"\" F-%$cosGF-\"\"#F-F'F-F-**&F+6#F/F-%$sinGF-F/F-F'F-F-" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 45 "The appropriate trial particular solu tion is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=p(t)" " 6#/%\"xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*exp(-t/2)*cos*t+B*exp(-t /2)*sin*t;" "6#/%!G,&**%\"AG\"\"\"-%$expG6#,$*&%\"tGF(\"\"#!\"\"F0F(%$ cosGF(F.F(F(**%\"BGF(-F*6#,$*&F.F(F/F0F0F(%$sinGF(F.F(F(" }{XPPEDIT 18 0 "``=exp(-t/2)*(A*cos*t+B*sin*t)" "6#/%!G*&-%$expG6#,$*&%\"tG\"\" \"\"\"#!\"\"F.F,,&*(%\"AGF,%$cosGF,F+F,F,*(%\"BGF,%$sinGF,F+F,F,F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\" %#dtG!\"\"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F' \"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*(A*cos*t+B*sin *t)+exp(-t/2)*(-A*sin*t+B*cos*t)" "6#,&*&-%$expG6#,$*&%\"tG\"\"\"\"\"# !\"\"F-F+,&*(%\"AGF+%$cosGF+F*F+F+*(%\"BGF+%$sinGF+F*F+F+F+F+*&-F&6#,$ *&F*F+F,F-F-F+,&*(F0F+F4F+F*F+F-*(F3F+F1F+F*F+F+F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(-t/2)*((B-A/ 2)*cos*t-(A+B/2)*sin*t)" "6#/%!G*&-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F. F,,&*(,&%\"BGF,*&%\"AGF,F-F.F.F,%$cosGF,F+F,F,*(,&F4F,*&F2F,F-F.F,F,%$ sinGF,F+F,F.F," }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) = `p \+ ''`(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+ " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*((B-A/2)*cos*t-(A+B/2)*sin*t) +exp(-t/ 2)*(-(B-A/2)*sin*t-(A+B/2)*cos*t)" "6#,&*&-%$expG6#,$*&%\"tG\"\"\"\"\" #!\"\"F-F+,&*(,&%\"BGF+*&%\"AGF+F,F-F-F+%$cosGF+F*F+F+*(,&F3F+*&F1F+F, F-F+F+%$sinGF+F*F+F-F+F+*&-F&6#,$*&F*F+F,F-F-F+,&*(,&F1F+*&F3F+F,F-F-F +F8F+F*F+F-*(,&F3F+*&F1F+F,F-F+F+F4F+F*F+F-F+F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(-t/2)*((A-3*B/4)*sin*t-(3*A/4+B)*cos*t);" "6#/ %!G*&-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F.F,,&*(,&%\"AGF,*(\"\"$F,%\"BG F,\"\"%F.F.F,%$sinGF,F+F,F,*(,&*(F4F,F2F,F6F.F,F5F,F,%$cosGF,F+F,F.F, " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " } {XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF %F'!\"\"" }{TEXT -1 5 " and " }{TEXT 321 1 "x" }{TEXT -1 37 " in the d ifferential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-t/2)*((A-3*B/4)*sin*t-(3*A/4+B)*cos*t)+4*exp(-t/2) *(A*cos*t+B*sin*t) = 5*exp(-t/2)*sin*t;" "6#/,&*&-%$expG6#,$*&%\"tG\" \"\"\"\"#!\"\"F.F,,&*(,&%\"AGF,*(\"\"$F,%\"BGF,\"\"%F.F.F,%$sinGF,F+F, F,*(,&*(F4F,F2F,F6F.F,F5F,F,%$cosGF,F+F,F.F,F,*(F6F,-F'6#,$*&F+F,F-F.F .F,,&*(F2F,F;F,F+F,F,*(F5F,F7F,F+F,F,F,F,**\"\"&F,-F'6#,$*&F+F,F-F.F.F ,F7F,F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Multiplying both sides by " }{XPPEDIT 18 0 "exp(t/2)" "6#-%$expG6#*&%\"tG\"\"\"\" \"#!\"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(A-3*B/4)*sin*t-(3*A/4+B)*cos*t+4*A*cos*t+4*B*sin*t = 5 *sin*t;" "6#/,**(,&%\"AG\"\"\"*(\"\"$F(%\"BGF(\"\"%!\"\"F-F(%$sinGF(% \"tGF(F(*(,&*(F*F(F'F(F,F-F(F+F(F(%$cosGF(F/F(F-**F,F(F'F(F3F(F/F(F(** F,F(F+F(F.F(F/F(F(*(\"\"&F(F.F(F/F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "By comparing the coeffi cients of the cosine terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "13*A/4-B = 0;" "6#/,&*(\"#8\"\"\"%\" AGF'\"\"%!\"\"F'%\"BGF*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "13*A-4*B = 0;" "6#/,&*&\"#8\"\"\"%\"AGF'F'*&\"\"%F'%\"BGF'!\"\" \"\"!" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "By comparing the coefficients of the si ne terms we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A+13*B/4 = 5;" "6#/,&%\"AG\"\"\"*(\"#8F&%\"BGF&\"\"%! \"\"F&\"\"&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*A+13*B = 20;" " 6#/,&*&\"\"%\"\"\"%\"AGF'F'*&\"#8F'%\"BGF'F'\"#?" }{TEXT -1 16 " ---- --- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Multiplying equation (i) by 4 and equation (ii) by 13 gives the pair of equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([52*A-16*B = 0, ``],[52*A+169*B = 260, ``]);" "6#-%*PIECE WISEG6$7$/,&*&\"#_\"\"\"%\"AGF+F+*&\"#;F+%\"BGF+!\"\"\"\"!%!G7$/,&*&F* F+F,F+F+*&\"$p\"F+F/F+F+\"$g#F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Subtracting the first of \+ these two equations from the second gives " }{XPPEDIT 18 0 "185*B=260 " "6#/*&\"$&=\"\"\"%\"BGF&\"$g#" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "B=260/185" "6#/%\"BG*&\"$g#\"\"\"\"$&=!\"\"" }{XPPEDIT 18 0 "``= 52/37" "6#/%!G*&\"#_\"\"\"\"#P!\"\"" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "A=4*B/13" "6#/%\"AG*(\"\"%\"\"\"%\"BGF'\"#8!\"\"" }{XPPEDIT 18 0 "``=16/37" "6#/%!G*&\"#;\"\"\"\"#P!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Thus we obtain th e particular solution " }{XPPEDIT 18 0 "p(t)=16/37" "6#/-%\"pG6#%\"tG *&\"#;\"\"\"\"#P!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*cos*t +52/37" "6#,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+%$cosGF+F*F+F+*& \"#_F+\"#PF-F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*sin*t" "6#*( -%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F,F*%$sinGF*F)F*" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "general solution" } {TEXT -1 47 " for the second differential equation given by " } {XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6#F'\"\"\"-%\" pG6#F'F," }{TEXT -1 14 " is therefore " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)= C[1]*cos*2*t+C[2]*sin*2*t+16/37" "6#/-%\"x G6#%\"tG,(**&%\"CG6#\"\"\"F-%$cosGF-\"\"#F-F'F-F-**&F+6#F/F-%$sinGF-F/ F-F'F-F-*&\"#;F-\"#P!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2) *cos*t+52/37" "6#,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+%$cosGF+F*F +F+*&\"#_F+\"#PF-F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*sin*t" "6#*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F,F*%$sinGF*F)F*" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Sinc e " }{XPPEDIT 18 0 "x(0) = 0;" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 25 ", we obtain the equation " }{XPPEDIT 18 0 "0 = C[1]+16/37;" "6#/\"\"!,&&% \"CG6#\"\"\"F)*&\"#;F)\"#P!\"\"F)" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "C[1] = -16/37;" "6#/&%\"CG6#\"\"\",$*&\"#;F'\"#P!\"\"F, " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-16/37" "6#/%\"xG,$*&\"#;\"\" \"\"#P!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*t+C[2]*sin*2*t+16 /37;" "6#,(*(%$cosG\"\"\"\"\"#F&%\"tGF&F&**&%\"CG6#F'F&%$sinGF&F'F&F(F &F&*&\"#;F&\"#P!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*cos* t+52/37" "6#,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+%$cosGF+F*F+F+*& \"#_F+\"#PF-F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*sin*t" "6#*( -%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F,F*%$sinGF*F)F*" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=-8/37" "6#/*&%#dxG\"\"\"%#dtG!\"\",$*&\"\")F &\"#PF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*t+2*C[2]*cos*2*t-8/37 " "6#,(*(%$sinG\"\"\"\"\"$F&%\"tGF&F&*,\"\"#F&&%\"CG6#F*F&%$cosGF&F*F& F(F&F&*&\"\")F&\"#P!\"\"F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)* cos*t-16/37" "6#,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+%$cosGF+F*F+ F+*&\"#;F+\"#PF-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*sin*t-26 /37" "6#,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+%$sinGF+F*F+F+*&\"#E F+\"#PF-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*sin*t+52/37" "6# ,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+%$sinGF+F*F+F+*&\"#_F+\"#PF- F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*cos*t" "6#*(-%$expG6#,$* &%\"tG\"\"\"\"\"#!\"\"F,F*%$cosGF*F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Eval(dx/dt,t=0)=0" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"! F." }{TEXT -1 17 ", it follws that " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "0=2*C[2]-8/37+52/37" "6#/\"\"!,(*&\"\"#\"\"\"&%\"CG6 #F'F(F(*&\"\")F(\"#P!\"\"F/*&\"#_F(F.F/F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "C[2] = -22/37;" "6#/&%\" CG6#\"\"#,$*&\"#A\"\"\"\"#P!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 259 19 "particular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=-16/37" "6# /-%\"xG6#%\"tG,$*&\"#;\"\"\"\"#P!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*t-22/37" "6#,&*(%$cosG\"\"\"\"\"#F&%\"tGF&F&*&\"#AF&\"#P!\"\" F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*t +16/37" "6#,&*(%$sinG\"\" \"\"\"#F&%\"tGF&F&*&\"#;F&\"#P!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*cos*t+52/37" "6#,&*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F-F+% $cosGF+F*F+F+*&\"#_F+\"#PF-F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/ 2)*sin*t" "6#*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F,F*%$sinGF*F)F*" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 322 37 "_____ ________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "satisfies the given initial condtions. " }}{PARA 0 "" 0 " " {TEXT -1 20 "In this example the " }{TEXT 259 14 "transient part" } {TEXT -1 17 " of the solution " }{XPPEDIT 18 0 "phi[trans](t) = 16/37; " "6#/-&%$phiG6#%&transG6#%\"tG*&\"#;\"\"\"\"#P!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/2)*cos*t+52/37;" "6#,&*(-%$expG6#,$*&%\"tG\"\" \"\"\"#!\"\"F-F+%$cosGF+F*F+F+*&\"#_F+\"#PF-F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "exp(-t/2)*sin*t;" "6#*(-%$expG6#,$*&%\"tG\"\"\"\"\"#!\" \"F,F*%$sinGF*F)F*" }{TEXT -1 128 " is the (first) particular solutio n based on the standard trial particular solution deduced from the for m of the forcing term. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 12 "steady-state" }{TEXT -1 23 " par t of the solution " }{XPPEDIT 18 0 "phi[sstate](t) = -16/37;" "6#/-&% $phiG6#%'sstateG6#%\"tG,$*&\"#;\"\"\"\"#P!\"\"F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*2*t-22/37;" "6#,&*(%$cosG\"\"\"\"\"#F&%\"tGF&F&*&\" #AF&\"#P!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*t;" "6#*(%$sinG \"\"\"\"\"#F%%\"tGF%" }{TEXT -1 135 " originates from the complementar y solution after the values of the constants as determined by the init ial conditions are substituted. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 265 1 "t" }{TEXT -1 26 " is \+ \"large\" the affect of " }{XPPEDIT 18 0 "p(t) = phi[trans](t);" "6#/- %\"pG6#%\"tG-&%$phiG6#%&transG6#F'" }{TEXT -1 110 " becomes neglible, \+ and the behaviour of the sytem is then essentially described by the co mplementary solution " }{XPPEDIT 18 0 "c(t) = phi[sstate](t);" "6#/-% \"cG6#%\"tG-&%$phiG6#%'sstateG6#F'" }{TEXT -1 114 ". The external for ce dies away and we are left with the free (undamped and unforced) osc illations of the system. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 40 "The graph of the solution is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "g := t -> -16/37*cos(2*t)-22/37*sin(2*t)+16/37*exp(-t/2)*cos(t)+52/37*e xp(-t/2)*sin(t):\n'g(t)'=g(t);\nplot(g(t),t=0..5*Pi,labels=[`t`,`x(t)` ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,**&#\"#;\"#P\"\" \"-%$cosG6#,$*&\"\"#F-F'F-F-F-!\"\"*&#\"#AF,F--%$sinGF0F-F4*&#F+F,F-*& -%$expG6#,$*&F3F4F'F-F4F--F/F&F-F-F-*&#\"#_F,F-*&F=F--F9F&F-F-F-" }} {PARA 13 "" 1 "" {GLPLOT2D 589 275 275 {PLOTDATA 2 "6%-%'CURVESG6$7]v7 $$\"\"!F)F(7$$\"39p!exSA(f&)!#>$\"3nM)=$p(>l5&!#@7$$\"3$Qh^:[W>r\"!#=$ \"35q#z](p6xR!#?7$$\"3u?uKAn\"zc#F4$\"3_As<0L&>I\"F-7$$\"3mFK5j*))QU$F 4$\"3yAyyzL=#)HF-7$$\"3<')Qy-FW8\\F4$\"3Q'=;mgZrA)F-7$$\"3pWXYUk*HS'F4 $\"3q[UGXt9z;F47$$\"3OSyH5a:y!)F4$\"3gR+Z![*yJIF47$$\"3)[8J\"yVJ`(*F4$ \"37M'[szEar%F47$$\"3(>'))pQx&R9\"!#<$\"3SV3#e\\+:f'F47$$\"3n5YeRSe78F Y$\"3WX%H/.7kW)F47$$\"3obt4*))3/[\"FY$\"3uQ].Wnw/5FY7$$\"3q+,hQPB[;FY$ \"3/nr]R!)*)=6FY7$$\"3D1t*yHJgs\"FY$\"3!Gww-Dq6FY7$$\"3O<yMF!\\%o6FY7$$\"3-?`6'>D0#>FY$\"33$*=t,#*Qi6FY7$$\"3!H#*ed (RUf>FY$\"3)>()R%Ro,_6FY7$$\"3>E;,kGFY$!3q]vF$Q\\L9'F-7$$\"3s F?!>Saq%HFY$!3AE??I\\B&4#F47$$\"3f(p]/6uC.$FY$!3v2,Nh05UNF47$$\"3Yn$** *=Q*y6$FY$!3#oQ6B#=4x[F47$$\"3*o.[v_8L?$FY$!3C'[(*orD#ogF47$$\"3w1n4OK t)G$FY$!3C1zg!>gr3(F47$$\"3^wem(o3#RMFY$!3C)eRjOvLR)F47$$\"3EY]BRTo*e$ FY$!3Sv\"=+,(o.!*F47$$\"3OC$[)oS.KOFY$!32o#=picR/*F47$$\"3!Hgh%)*RQuOF Y$!3B#4-)[tCE!*F47$$\"3X\")[2GRt;PFY$!3%HbEWNp5&*)F47$$\"3cf\")odQ3fPF Y$!3@3AokIJ>))F47$$\"3@;Z\"pr$yVQFY$!3w,GHL3\\\"R)F47$$\"3'GFThd$[GRFY $!31GKwx:AdxF47$$\"324w7@7`8SFY$!3d9)oV]lE$pF47$$\"3%[%R6m)y&)4%FY$!3] (f`n(o5XfF47$$\"3g!G+6^EO=%FY$!3([D#yQs\"R#[F47$$\"3Q;m3cTnoUFY$!3#3LA WaD>g$F47$$\"39Yj(4KL1N%FY$!3sa'GZhw>O#F47$$\"3!f2me[#fKWFY$!3wmx\"*e* yU4\"F47$$\"3n0ev];b9XFY$\"3=[*[%H$\\mn\"F-7$$\"3aMbk:3^'f%FY$\"3_>uiVOXZFY$\"3^]-Tv*=0T$F47$$\"3]/$Q)4z@%*[FY$\"3=xA8i a!z-&F47$$\"35m()HU&=F)\\FY$\"3+%*z?rj;UdF47$$\"3yE#fZ<>72&FY$\"3[d)3) R8%eC'F47$$\"39d%*)4\\pa6&FY$\"3!))4%\\DC?9kF47$$\"3[(o>s!)>(f^FY$\"3M ^')>[^QDlF47$$\"3s=*\\M7qR?&FY$\"3us7c.r!)ylF47$$\"31\\,oR/A[_FY$\"3A% 4?w`TVd'F47$$\"3%HJi$GP;)R&FY$\"3!>XeZO%yOhF47$$\"3'fZWq,2\"[bFY$\"3Qt 2\"z^kL4&F47$$\"3w#>_B'oJNcFY$\"31,S]Rj?ZUF47$$\"3c4*fwqEDs&FY$\"3QooZ jW[eKF47$$\"3PEw'HbO(4eFY$\"3'3I!z*oS$f@F47$$\"3/W`F)RYp*eFY$\"3S5cMl' R>&)*F-7$$\"37!G7&))=9ufFY$!3')yGR#\\obl)F77$$\"3I:#\\(ytL^gFY$!3ChP&= 883;\"F47$$\"3Y]h)*oG`GhFY$!3XYj;cqx5AF47$$\"3j&3B#f$Gd?'FY$!3?-I;,GH5 KF47$$\"3)e(pXM(>^P'FY$!3o7>c=Qe2^F47$$\"38m3p46^WlFY$!3%f1ar[4!>kF47$ $\"3(pg(3L9;DmFY$!3McHmt$y')y'F47$$\"3!yM%[c<\"eq'FY$!3#yZ>xnY\"ypF47$ $\"3A=F==p8YnFY$!3rK;aU3O.qF47$$\"3i)3\"))z?Y'y'FY$!3]b,VqF!=)pF47$$\" 3;e%z:C(yEoFY$!3'H8.B%*RN\"pF47$$\"3cGyF.C6noFY$!3Kw!z%e*[*)z'F47$$\"3 Q.yl1FTNqFY$!3\"[Vt\\Ix4%eF47$$\"34zx.5Ir.sFY$!3%[s)os@8*>%F47$$\"3+MN 7=\"*)4G(FY$!3gEE^?4WAF47$$\"30U]HM 8aNuFY$!3Sp6!\\.xU;\"F47$$\"3'pz!QUu\"G^(FY$!3nTq$3<&y4]F77$$\"3!z'3fk ,<'f(FY$\"3#)e\"p'Qgff6F47$$\"3')Q4!o)G_zwFY$\"3*p_feZXYM#F47$$\"3#)45 ,4c(Gw(FY$\"3uLwD7.-sMF47$$\"3w!3@7LGi%yFY$\"3P\\]qxAC5XF47$$\"3r%*>EO %)Q>!)FY$\"3;[gg%)=qfiF47$$\"3U5HIT&[D>)FY$\"3WS`Cd$yBH(F47$$\"3+\"G@k jK-B)FY$\"3Zq438\"ozS(F47$$\"3c^'R:t;zE)FY$\"3G-vmu_d#[(F47$$\"38A!em# 3g0$)FY$\"3\\%3pClRd^(F47$$\"3p#Rw<#\\GV$)FY$\"3zYg,'*eB2vF47$$\"3#Q88 ?6`'=%)FY$\"3?)\\7kf.bO(F47$$\"3%\\()\\AI@S\\)FY$\"3-kk[#\\!HgqF47$$\" 3e?$>Fs?ol)FY$\"3SX%\\>b_#zeF47$$\"3>m()=V,i>))FY$\"3c^S)Q_cC4%F47$$\" 3`6'G)3Sr.*)FY$\"3#=1S)=r\"4*HF47$$\"3ke%oW(y!y)*)FY$\"3@2H>O2-2=F47$$ \"3u0$3,u,>2*FY$\"3F\"GsW\\#*zt&F-7$$\"3'G:[dg&*f:*FY$!3q:L6s72VnF-7$$ \"30\\/4Z]EQ#*FY$!3;xC!yWIi(=F47$$\"3CXFV)[M0K*FY$!3Q**fTtR&o-$F47$$\" 3?V]xHR!GS*FY$!3#pp&*))*=Q&4%F47$$\"3SRt6rL2&[*FY$!39mG*=?;K0&F47$$\"3 #4SY0?tUk*FY$!3EosSyT3+lF47$$\"3@ka(*HIZ.)*FY$!3K[fAB1r&H(F47$$\"3s,YW j&ob#)*FY$!3U;]qntO\\tF47$$\"3ARP\"p4kw%)*FY$!3O#>J2BW()Q(F47$$\"3uwGQ I'f(p)*FY$!3i44&=:nPT(F47$$\"3C9?&Q;b=*)*FY$!3Qv@[\"***QCuF47$$\"3E*G! zIi/O**FY$!3R;%4A'4R-uF47$$\"3Gk&GxHP-)**FY$!3T?I)e_QHK(F47$$\"3V60;V> 'o+\"!#;$!3e@;l'HoY*pF47$$\"3Em\"[l:+d,\"Fb^m$!3i4JSuU$*\\kF47$$\"3[Bm >7KeJ5Fb^m$!3!)4+f\"*3A&)\\F47$$\"3_!3XyEmu/\"Fb^m$!370!Gyb]s-$F47$$\" 3U=E)QaXf0\"Fb^m$!3=YftWMQY=F47$$\"37c,#*>[Uk5Fb^m$!3bGVQ_Y(>9'F-7$$\" 3%Qpdf4/H2\"Fb^m$\"3#3\"\\,+P$Q\"3\"Fb^m$\"3;4uR7]Ki= F47$$\"3OdMNco1*3\"Fb^m$\"3QFI6zS\")GHF47$$\"3)Ho62M]n4\"Fb^m$\"3h<1F \\H,DRF47$$\"3W3*p]#QV/6Fb^m$\"3z%f<5i%[F[F47$$\"32M\"G%4t676Fb^m$\"3. 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t):\ng := unapply(phi[trans](t)+phi[sstate](t),t):\n'phi[trans](t)'=ph i[trans](t);\n'phi[sstate](t)'=phi[sstate](t);\n'g(t)'=g(t);\nplot([g( t),phi[trans](t),phi[sstate](t)],t=0..3*Pi,color=[red,blue,magenta],\n legend=[`complete solution`,`transient part of solution`,`steady-sta te part of solution`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$phiG6# %&transG6#%\"tG,&*&#\"#;\"#P\"\"\"*&-%$expG6#,$*&\"\"#!\"\"F*F0F8F0-%$ cosGF)F0F0F0*&#\"#_F/F0*&F2F0-%$sinGF)F0F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$phiG6#%'sstateG6#%\"tG,&*&#\"#;\"#P\"\"\"-%$cosG6# ,$*&\"\"#F0F*F0F0F0!\"\"*&#\"#AF/F0-%$sinGF3F0F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG,**&#\"#;\"#P\"\"\"*&-%$expG6#,$*&\"\"#! \"\"F'F-F5F--%$cosGF&F-F-F-*&#\"#_F,F-*&F/F--%$sinGF&F-F-F-*&#F+F,F--F 76#,$*&F4F-F'F-F-F-F5*&#\"#AF,F--F=FAF-F5" }}{PARA 13 "" 1 "" {GLPLOT2D 614 326 326 {PLOTDATA 2 "6'-%'CURVESG6%7[s7$$\"\"!F)F(7$$\"3 \"zBM*)omr-\"!#=$\"3Kl9-2fUz()!#@7$$\"3#eZoyPLV0#F-$\"3W.8nLi9#z'!#?7$ 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e`)fF-7$Fjil$!3sP%o5bKq-&F-7$F_jl$!3%fZ9_b5<(QF-7$Fdjl$!39l4P'*>8&f#F- 7$Fijl$!38I\"p.rJ8A\"F-7$$\"37U7R!yuo](F]o$!3XO'pVxah![F<7$F^[m$\"3)[Q )GO_X]EF<7$$\"310i%H5]$3wF]o$\"3h+J?G'zz+\"F-7$Fc[m$\"3>)Q%3FKaSrJkZ4$F-7$F]\\m$\"35d%y4O?6L%F-7$Fb\\m$\"3:>4=iLaCaF-7$Fg\\ m$\"3ibY_(3S>I'F-7$F\\]m$\"3#G(H9kyGCpF-7$Fa]m$\"3')\\64jmTvsF-7$Ff]m$ \"3o1AhjF5;tF-7$F[^m$\"3\\[#G[()=;M(F-7$F`^m$\"3E1v!*)37>N(F-7$Fe^m$\" 3M>**zi5'pM(F-7$Fj^m$\"3+8@\\igxEtF-7$F__m$\"3i$z?<%*)R\"H(F-7$Fd_m$\" 3+T()=bI!4C(F-7$Fi_m$\"3;RLY1JRvrF-7$F^`m$\"3Gv$zL#>Y(o'F-7$Fc`m$\"3X( f#4$o-\"4fF-7$Fh`m$\"3a1;W)GhU*\\F-7$F]am$\"3\\W?(*RSy0RF-7$Fbam$\"3G3 &[W#pn'f#F-7$Fgam$\"3a75W[:)\\=\"F-7$F\\bm$!35;TQRg33@F<7$Fabm$!3GVz&[ $p'*)f\"F-7$Ffbm$!3YgK0Oy%[-$F-7$F[cm$!3;\\%)G+TKCVF--F`cm6&FbcmFccmF( Fccm-Fgcm6#%>steady-state~part~of~solutionG-%+AXESLABELSG6$Q\"t6\"Q!Fe `p-%%VIEWG6$;F($\"+izxC%*!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "complete solution" "transient part o f solution" "steady-state part of solution" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Solution by " }{TEXT 0 6 "dsolve" }{TEXT -1 5 " and " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "de := diff(x (t),t$2)+4*x(t)=5*exp(-t/2)*sin(t);\nic := x(0)=0,D(x)(0)=0;\ndsolve( \{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6 $-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,$*(\"\"&F2-%$expG6# ,$*&F1!\"\"F-F2F=F2-%$sinGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&#\"#A\"#P\"\"\"-%$sinG6#,$*&\"\"#F-F'F-F-F-!\" \"*&#\"#;F,F--%$cosGF0F-F4*&#F-F,F-*&,&*&\"#_F--F/F&F-F-*&F7F--F9F&F-F -F--%$expG6#,$*&F3F4F'F-F4F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "desolve" } {TEXT -1 24 " can be used instead of " }{TEXT 0 6 "dsolve" }{TEXT -1 45 " if we want to see the steps in the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "de := diff(x (t),t$2)+4*x(t)=5*exp(-t/2)*sin(t);\nic := x(0)=0,D(x)(0)=0;\ndesolve( \{de,ic\},info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%d iffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,$*(\"\"&F2-%$e xpG6#,$*&F1!\"\"F-F2F=F2-%$sinGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F*\"\"% F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G^#\"\"#^#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F,F,F,F -F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)~~~~~~~=G,$*&#\"\"%\"#P\"\" \"*&-%$expG6#,$*&\"\"#!\"\"%\"tGF)F1F),&*&\"#8F)-%$sinG6#F2F)F)*&F'F)- %$cosGF8F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6general~solution~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&#\"\"%\"#P\"\"\"*&-%$expG6#,$*&\"\"#! \"\"F'F-F5F-,&*&\"#8F--%$sinGF&F-F-*&F+F--%$cosGF&F-F-F-F-F-*&&%\"CG6# F-F--F:6#,$*&F4F-F'F-F-F-F-*&&F@6#F4F--F=FCF-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afrom~the~initi al~conditions~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!,&#\"#;\" #P\"\"\"&%\"CG6#\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!,&#\" #W\"#P\"\"\"*&\"\"#F)&%\"CG6#F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.~.~G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"#!#A\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"##!#;\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&#\"\"%\"#P \"\"\"*&-%$expG6#,$*&\"\"#!\"\"F'F-F5F-,&*&\"#8F--%$sinGF&F-F-*&F+F--% $cosGF&F-F-F-F-F-*&#\"#AF,F--F:6#,$*&F4F-F'F-F-F-F5*&#\"#;F,F--F=FBF-F 5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Resonance .. introductory example " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Consider t he initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+9*x = 5*cos*3*t;" "6#/,&*(%\"dG\"\"#%\"xG \"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"\"*F)F(F)F)**\"\"&F)%$cosGF)\"\"$F)F ,F)" }{TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) \+ = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 94 "which involves a system with no dam ping, but with an external force which varies sinusoidally." }{TEXT 261 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 22 "complementary solution" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "c(t) = C[1]*cos*3*t+C[2]*sin*3*t;" "6#/-%\"cG6#%\"tG,&* *&%\"CG6#\"\"\"F-%$cosGF-\"\"$F-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The \"forcing function\" " }{XPPEDIT 18 0 "5*cos*3*t;" "6 #**\"\"&\"\"\"%$cosGF%\"\"$F%%\"tGF%" }{TEXT -1 138 " duplicates one o f the functions which appear in the complementary solution. Hence the \+ appropriate form for a trial particular solution is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" } {XPPEDIT 18 0 "`` = A*t*cos*3*t+B*t*sin*3*t;" "6#/%!G,&*,%\"AG\"\"\"% \"tGF(%$cosGF(\"\"$F(F)F(F(*,%\"BGF(F)F(%$sinGF(F+F(F)F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\" \"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*3*t-3*A*t*sin*3*t+B*sin*3 *t+3*B*t*cos*3*t;" "6#/%!G,***%\"AG\"\"\"%$cosGF(\"\"$F(%\"tGF(F(*.F*F (F'F(F+F(%$sinGF(F*F(F+F(!\"\"**%\"BGF(F-F(F*F(F+F(F(*.F*F(F0F(F+F(F)F (F*F(F+F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) = `p '' `(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -6*A*sin*3*t-9*A*t*cos*3*t+6*B*cos*3*t-9*B*t*sin* 3*t;" "6#/%!G,**,\"\"'\"\"\"%\"AGF(%$sinGF(\"\"$F(%\"tGF(!\"\"*.\"\"*F (F)F(F,F(%$cosGF(F+F(F,F(F-*,F'F(%\"BGF(F0F(F+F(F,F(F(*.F/F(F2F(F,F(F* F(F+F(F,F(F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Substitut ing for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F $F'*$%\"tGF%F'!\"\"" }{TEXT -1 5 " and " }{TEXT 274 1 "x" }{TEXT -1 37 " in the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-6*A*sin*3*t-9*A*t*cos*3*t+6*B*cos*3*t-9*B*t* sin*3*t +9*A*t*cos*3*t+9*B*t*sin*3*t = 5*cos*3*t" "6#/,.*,\"\"'\"\"\"% \"AGF'%$sinGF'\"\"$F'%\"tGF'!\"\"*.\"\"*F'F(F'F+F'%$cosGF'F*F'F+F'F,*, F&F'%\"BGF'F/F'F*F'F+F'F'*.F.F'F1F'F+F'F)F'F*F'F+F'F,*.F.F'F(F'F+F'F/F 'F*F'F+F'F'*.F.F'F1F'F+F'F)F'F*F'F+F'F'**\"\"&F'F/F'F*F'F+F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-6*A*sin*3*t+6*B*cos*3*t = 5*cos*3*t;" "6#/,&*,\"\"'\"\"\"%\"AGF'%$sinGF'\"\"$F'%\"tGF'!\"\"*,F&F'%\"BGF'%$co sGF'F*F'F+F'F'**\"\"&F'F/F'F*F'F+F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 21 "Hence we should take " }{XPPEDIT 18 0 "A=0" "6#/%\"AG\" \"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B = 5/6;" "6#/%\"BG*&\"\"&\" \"\"\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Thus w e obtain the particular solution " }{XPPEDIT 18 0 "p(t) = 5/6;" "6#/- %\"pG6#%\"tG*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t *sin*3*t" "6#**%\"tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "general solution" } {TEXT -1 2 " " }{XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-% \"cG6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 44 " for the differential equati on is therefore " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(t) = C[1]*cos*3 *t+C[2]*sin*3*t+5/6;" "6#/-%\"xG6#%\"tG,(**&%\"CG6#\"\"\"F-%$cosGF-\" \"$F-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-*&\"\"&F-\"\"'!\"\"F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**%\"tG\"\"\"%$sinGF%\"\" $F%F$F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straigh tforward to check that the " }{TEXT 259 19 "particular solution" } {TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t ) = 5/6;" "6#/-%\"xG6#%\"tG*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "t*sin*3*t" "6#**%\"tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 275 10 "__________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "obtained by taking " }{XPPEDIT 18 0 "C[1]=C[2]" "6#/&%\"CG6#\"\"\"&F%6#\"\"#" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 41 ", satisfies the given initial con dtions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de := diff(x(t),t$2)+ 9*x(t)=5*cos(3*t);\nic := x(0)=0,D(x)(0)=0;\ndsolve(\{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tG-%\" $G6$F-\"\"#\"\"\"*&\"\"*F2F*F2F2,$*&\"\"&F2-%$cosG6#,$*&\"\"$F2F-F2F2F 2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"D G6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*&#\"\"& \"\"'\"\"\"*&-%$sinG6#,$*&\"\"$F-F'F-F-F-F'F-F-F-" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The factor " }{TEXT 276 1 "t" }{TEXT -1 28 " ap pearing in the solution " }{XPPEDIT 18 0 "x = 5/6;" "6#/%\"xG*&\"\"& \"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**%\"t G\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 46 " means that the amplitude of the oscillations " }{TEXT 259 22 "increases indefinitely" }{TEXT -1 4 " as " }{TEXT 277 1 "t" }{TEXT -1 11 " increases." }}{PARA 0 "" 0 " " {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "x = 5/6;" "6#/%\"xG*&\" \"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**% \"tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 47 " lies entirely between t he two straight lines " }{XPPEDIT 18 0 "x=5/6*t" "6#/%\"xG*(\"\"&\"\" \"\"\"'!\"\"%\"tGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = -5/6*t;" "6#/%\"xG,$*(\"\"&\"\"\"\"\"'!\"\"%\"tGF(F*" }{TEXT -1 2 ". 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Resonance in me chanical structures can lead to catastrophic failure." }}{PARA 0 "" 0 "" {TEXT -1 147 "Resonance is not always destructive. For example, it \+ is resonance in an electronic circuit which enables a radio to be tune d to a specific station." }}{PARA 0 "" 0 "" {TEXT -1 142 "In practice \+ resonance of the kind described here, with an unlimitted increase in a mplitude, does not occur because of the presence of damping." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " } {XPPEDIT 18 0 "x = 5/6;" "6#/%\"xG*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**%\"tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 16 " meets the line " }{XPPEDIT 18 0 "x=5*t/6" "6#/%\"xG*(\" \"&\"\"\"%\"tGF'\"\"'!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "sin*3 *t=1" "6#/*(%$sinG\"\"\"\"\"$F&%\"tGF&F&" }{TEXT -1 16 ", that is, whe n " }{XPPEDIT 18 0 "3*t = Pi/2,5*Pi/2,9*Pi/2,` . . . `;" "6&/*&\"\"$\" \"\"%\"tGF&*&%#PiGF&\"\"#!\"\"*(\"\"&F&F)F&F*F+*(\"\"*F&F)F&F*F+%(~.~. ~.~G" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "t = Pi/6, 5*Pi/6, 3*Pi/2, ` \+ . . . `" "6&/%\"tG*&%#PiG\"\"\"\"\"'!\"\"*(\"\"&F'F&F'F(F)*(\"\"$F'F&F '\"\"#F)%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "At these values of " }{TEXT 278 1 "t" }{TEXT -1 17 ", the derivative " } {XPPEDIT 18 0 "dx/dt=5/6" "6#/*&%#dxG\"\"\"%#dtG!\"\"*&\"\"&F&\"\"'F( " }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*t+5/2" "6#,&*(%$sinG\"\"\"\"\" $F&%\"tGF&F&*&\"\"&F&\"\"#!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*c os*3*t" "6#**%\"tG\"\"\"%$cosGF%\"\"$F%F$F%" }{TEXT -1 13 " is equal t o " }{XPPEDIT 18 0 "5/6" "6#*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 27 ". T his shows that the line " }{XPPEDIT 18 0 "x=5*t/6" "6#/%\"xG*(\"\"&\" \"\"%\"tGF'\"\"'!\"\"" }{TEXT -1 4 " is " }{TEXT 259 23 "tangential to the curve" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x = 5/6;" "6#/%\"xG*&\"\" &\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**%\" tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 48 " at the points correspondin g to these values of " }{TEXT 279 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " } {XPPEDIT 18 0 "x = 5/6;" "6#/%\"xG*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**%\"tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 16 " meets the line " }{XPPEDIT 18 0 "x = -5*t/6;" "6#/%\"xG ,$*(\"\"&\"\"\"%\"tGF(\"\"'!\"\"F+" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "sin*3*t = -1;" "6#/*(%$sinG\"\"\"\"\"$F&%\"tGF&,$F&!\"\"" }{TEXT -1 16 ", that is, when " }{XPPEDIT 18 0 "3*t = 3*Pi/2,7*Pi/2,11*Pi/2,` . . . `;" "6&/*&\"\"$\"\"\"%\"tGF&*(F%F&%#PiGF&\"\"#!\"\"*(\"\"(F&F)F &F*F+*(\"#6F&F)F&F*F+%(~.~.~.~G" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "t = Pi/2,7*Pi/6,11*Pi/6,` . . . `;" "6&/%\"tG*&%#PiG\"\"\"\"\"#!\"\"*( \"\"(F'F&F'\"\"'F)*(\"#6F'F&F'F,F)%(~.~.~.~G" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 19 "At these values of " }{TEXT 281 1 "t" } {TEXT -1 17 ", the derivative " }{XPPEDIT 18 0 "dx/dt=5/6" "6#/*&%#dxG \"\"\"%#dtG!\"\"*&\"\"&F&\"\"'F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin* 3*t+5/2" "6#,&*(%$sinG\"\"\"\"\"$F&%\"tGF&F&*&\"\"&F&\"\"#!\"\"F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "t*cos*3*t" "6#**%\"tG\"\"\"%$cosGF%\"\" $F%F$F%" }{TEXT -1 13 " is equal to " }{XPPEDIT 18 0 "-5/6;" "6#,$*&\" \"&\"\"\"\"\"'!\"\"F(" }{TEXT -1 27 ". This shows that the line " } {XPPEDIT 18 0 "x = -5*t/6;" "6#/%\"xG,$*(\"\"&\"\"\"%\"tGF(\"\"'!\"\"F +" }{TEXT -1 4 " is " }{TEXT 259 23 "tangential to the curve" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x = 5/6;" "6#/%\"xG*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6#**%\"tG\"\"\"%$sinGF%\" \"$F%F$F%" }{TEXT -1 48 " at the points corresponding to these values \+ of " }{TEXT 280 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot([5/6*t*sin(3*t),5/ 6*t,-5/6*t],t=0..4,color=[red,black$2],\n linestyle=[1,3,3],labels= [`t`,`x(t)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 " 6'-%'CURVESG6%7^p7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3wQaybr')y=F-7$$\" 3RLLLe'40j\"!#=$\"3Ar?FD\"4XQ'F-7$$\"3mmmm;6m$[#F3$\"3oM8+*yfLS\"F37$$ \"3fmmm;yYULF3$\"3yk,T@+%zM#F37$$\"3%HLL$eF>(>%F3$\"3\"3hT/!p=HLF37$$ \"3Qmmm\">K'*)\\F3$\"3_3BB;gnYTF37$$\"3P*****\\Kd,\"eF3$\"3%G_9rCY,x%F 37$$\"39KLLe9XMiF3$\"3c=8;n&RS'\\F37$$\"3-mmm\"fX(emF3$\"3A=ib)HB60&F3 7$$\"3_KLL3!z;3(F3$\"38Ne%=C<'>]F37$$\"3.*****\\U7Y](F3$\"3W3M4@v]g[F3 7$$\"3Wmm;H9lRzF3$\"3u=cd#>6nb%F37$$\"3'QLLLV!pu$)F3$\"3OpyUbJ*p5%F37$ $\"3xmmm;c0T\"*F3$\"3K=hwE:LhHF37$$\"3#*******H,Q+5!#<$\"3]c2i9Y.n6F37 $$\"3)*******\\*3q3\"F_p$!3E3!e4T8$z5F37$$\"3)*******p=\\q6F_p$!3&zwwI Foh_$F37$$\"3mmm;fBIY7F_p$!33iNrUBDTeF37$$\"3GLLLj$[kL\"F_p$!3!H(*zaKA 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ObbI*z7&QF_p7$Faw$!3!)****\\(3&z7?F_p7$Ffw$!3u***\\Pfh.3#F_p7$F`x$!3Qm m;zEQb@F_p7$Fdy$!3<666hCzAAF_p7$Fhz$!3_mmm\"zmZH#F_p7$Fb[l$!3u)))QEB') *fBF_p7$Fg[l$!3ULLL3$)GJCF_p7$Fa\\l$!3%36h[+y$)\\#F_p7$F[]l$!35++vog]o DF_p7$Fe]l$!3IWWWp!pqj#F_p7$F_^l$!3ZLL3x\"[)3FF_p7$Fi^l$!3?*)))))))*zz x#F_p7$Fc_l$!3s))))Q^sn[GF_p7$F]`l$!3?66'[5*y=HF_p7$Fb`l$!3)ommmh9K)HF _p7$F\\al$!3;66hBN0dIF_p7$Fjbl$!3KAAAAf4BJF_p7$Fdcl$!31++v$*3^$>$F_p7$ F^dl$!33++D1q!4E$F_p7$Fhdl$!3[LLLLLLLLF_pFj^mF\\_m-%+AXESLABELSG6$%\"t G%%x(t)G-%%VIEWG6$;F(Fhdl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Beats .. introductory example" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 143 "When the natural frequency of a system and the frequency of an applied forcing function differ by a relatively small amount, the pho nomenon of " }{TEXT 259 5 "beats" }{TEXT -1 11 " may occur." }}{PARA 0 "" 0 "" {TEXT -1 47 "For example, consider the initial value problem " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+1 21*x = 48*cos*13*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\" \"F)*&\"$@\"F)F(F)F)**\"#[F)%$cosGF)\"#8F)F,F)" }{TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG \"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 58 "related to a physical system with a natural frequency of \+ " }{XPPEDIT 18 0 "11/(2*Pi);" "6#*&\"#6\"\"\"*&\"\"#F%%#PiGF%!\"\"" } {TEXT -1 69 ", which experiences the action of a forcing function with frequency " }{XPPEDIT 18 0 "13/(2*Pi);" "6#*&\"#8\"\"\"*&\"\"#F%%#Pi GF%!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 259 22 "complementary solution" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "c(t) = C[1]*cos*11*t+C[2]*sin*11*t; " "6#/-%\"cG6#%\"tG,&**&%\"CG6#\"\"\"F-%$cosGF-\"#6F-F'F-F-**&F+6#\"\" #F-%$sinGF-F/F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "The appropriate form for a trial part icular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*13*t+B *sin*13*t;" "6#/%!G,&**%\"AG\"\"\"%$cosGF(\"#8F(%\"tGF(F(**%\"BGF(%$si nGF(F*F(F+F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "`` = -13*A* sin*13*t+13*B*cos*13*t;" "6#/%!G,&*,\"#8\"\"\"%\"AGF(%$sinGF(F'F(%\"tG F(!\"\"*,F'F(%\"BGF(%$cosGF(F'F(F+F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) = `p ''`(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$% \"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -169*A*cos*13*t-169*B* sin*13*t;" "6#/%!G,&*,\"$p\"\"\"\"%\"AGF(%$cosGF(\"#8F(%\"tGF(!\"\"*,F 'F(%\"BGF(%$sinGF(F+F(F,F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(% \"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 5 " and " } {TEXT 282 1 "x" }{TEXT -1 37 " in the differential equation gives: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-169*A*cos*13*t-169* B*sin*13*t+121*A*cos*13*t+121*B*sin*13*t = 48*cos*13*t;" "6#/,**,\"$p \"\"\"\"%\"AGF'%$cosGF'\"#8F'%\"tGF'!\"\"*,F&F'%\"BGF'%$sinGF'F*F'F+F' F,*,\"$@\"F'F(F'F)F'F*F'F+F'F'*,F1F'F.F'F/F'F*F'F+F'F'**\"#[F'F)F'F*F' F+F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 80 "By comparing coefficients of the cosine and sine terms \+ we obtain the equations: " }{XPPEDIT 18 0 "-48*A = 48;" "6#/,$*&\"#[\" \"\"%\"AGF'!\"\"F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-48*B = 0;" "6 #/,$*&\"#[\"\"\"%\"BGF'!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "A = -1;" "6#/%\"AG,$\"\"\"!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B=0" "6#/%\"BG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular soluti on " }{XPPEDIT 18 0 "p(t) = -cos*13*t;" "6#/-%\"pG6#%\"tG,$*(%$cosG\" \"\"\"#8F+F'F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 259 16 "general solution" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x (t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," } {TEXT -1 44 " for the differential equation is therefore " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(t) = C[1]*co s*11*t+C[2]*sin*11*t-cos*13*t;" "6#/-%\"xG6#%\"tG,(**&%\"CG6#\"\"\"F-% $cosGF-\"#6F-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-*(F.F-\"#8F-F'F-!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straightforw ard to check that the " }{TEXT 259 19 "particular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = cos*1 1*t-cos*13*t;" "6#/-%\"xG6#%\"tG,&*(%$cosG\"\"\"\"#6F+F'F+F+*(F*F+\"#8 F+F'F+!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "satisfies \+ the given initial condtions. " }}{PARA 0 "" 0 "" {TEXT -1 79 "By means of a standard trig. formula, the solution can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 2*sin*t*s in*12*t" "6#/-%\"xG6#%\"tG*.\"\"#\"\"\"%$sinGF*F'F*F+F*\"#7F*F'F*" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 283 12 "____ ________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "de : = diff(x(t),t$2)+121*x(t)=48*cos(13*t);\nic := x(0)=0,D(x)(0)=0;\ndeso lve(\{de,ic\},x(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&\"$@\"F2F*F2F2,$-%$ cosG6#,$F-\"#8\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6 #\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliar y~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F*\"$@\"F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~. ~G^#\"#6^#!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%\"gGf*6#%\"tG6\"6$%)operatorG% &arrowGF(,&-%$cosG6#,$9$\"#8!\"\"-F.6#,$F1\"#6\"\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "We can use " } {TEXT 0 16 "combine(..,trig)" }{TEXT -1 56 " to check that the solutio n can be written in the form " }{XPPEDIT 18 0 "g(t) = 2*sin*t*sin*12* t;" "6#/-%\"gG6#%\"tG*.\"\"#\"\"\"%$sinGF*F'F*F+F*\"#7F*F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "combine(2*sin(t)*sin(12*t),trig);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&-%$cosG6#,$%\"tG\"#8!\"\"-F%6#,$F(\"#6\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Plotting the graph of the solution illustrates the phonomenon of beats in whic h the amplitude of the oscillations varies in a sinusoidal manner." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(2*sin(t)*sin(12*t),t=0..3*Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 154 154 {PLOTDATA 2 "6%-%'CURVESG6$7i]m7$$\"\"!F)F(7$$\"3_*=rYWLe8 &!#>$\"3[qnhCwhMfF-7$$\"3\"zBM*)omr-\"!#=$\"3Y^B;2*eX$>F37$$\"3Q(zn6Oe RG\"F3$\"39UP476rfDF37$$\"3'oN,M.]2a\"F3$\"3')zD]sAR^HF37$$\"3gO\"=&pe 9p;F3$\"3/$*3uRuHF3$\"3C_\"o/w(3EGF37$$\"3#eZoyPLV0#F3$\"3g?\\8)z1Rb#F37$$\"3]&4u(*\\ *>,DF3$\"3eH`E/(fa\"pF-7$$\"3?:(z;il![HF3$!3Lr\\q46)>C#F37$$\"3WN`eV<$ \\R$F3$!3KLs)*=;=[`F37$$\"39b4\\lyzTQF3$!3%o_tnbRoX(F37$$\"3A.\\yEjh/R F3$!3J@I60Kc4wF37$$\"3'=&)y!)yMu'RF3$!3ha#eHdO#>xF37$$\"3\\+GP\\KDISF3 $!3kD')f)*e\"Ry(F37$$\"37\\nm5<2$4%F3$!3m5RD+l#>!yF37$$\"3SYYDL'3(=UF3 $!3sJE:-!*o#p(F37$$\"3nVD%ebXVM%F3$!3e(**ej=7TQ(F37$$\"3kP$=5S>cf%F3$! 3qm:)R7jb;'F37$$\"3kJT>YK*o%[F3$!37WB7.Oj%>%F37$$\"31D*p84n\")4&F3$!3) p:q/7kog\"F37$$\"3i>daO4W\\`F3$\"3S\"*[%3+>QQ\"F37$$\"3;9:s\"y92g&F3$ \"3XGm&f>#>.XF37$$\"3g2t*oi))>&eF3$\"31UI6C7Fhs7$$\"3G[zrV=/ umF3$\"39U*o$yI3B7Fhs7$$\"3'eI]lqwst'F3$\"3i^/`>Rt97Fhs7$$\"3VjEQp:^+o F3$\"3[&)H\"fX-!*>\"Fhs7$$\"3!*>]@KkujoF3$\"3!\\$p#=ase<\"Fhs7$$\"3)zW WN)eo;rF3$\"3U(Q\\7G#o55Fhs7$$\"31wQ([LD'ptF3$\"3m#\\]Xp%pztF37$$\"33! fQ01&4'\\(F3$\"3]t:ea>3ucF37$$\"38/L?'ykDi(F3$\"3Af&)RV8C'y$F37$$\"3;= !o=^M!\\xF3$\"31C5QE$ovv\"F37$$\"3AKF`PU]vyF3$!3Al-=^Ts7)F3$!3uXz&Q K3!*F3$!3r`I,UuwS:Fhs7$$\"3fEB))31>M\"*F3$!3io'*3:0D#e\"Fhs7$$\"3!o!) \\u#\\7(>*F3$!3Y/An\"[>&*e\"Fhs7$$\"3#fGw*ph1A(F37$$\"3-qs?[p'R4\"Fhs$\"3-LV'>L[sX*F37$$\"3&zQ+@eOc5\"F hs$\"3y3)ot9sK:\"Fhs7$$\"3)e]$*f@1t6\"Fhs$\"3/p,L$4=.M\"Fhs7$$\"3.Cm)) \\e(*G6Fhs$\"3p0$\\Q,XG]\"Fhs7$$\"35gGn<^J_6Fhs$\"3:GoVz-lS\"Fhs$\"3t$oJ9;t4!=Fhs7$ $\"3v!4=`I()RA\"Fhs$\"3\"HEDXSkGg\"Fhs7$$\"3hRGGN02O7Fhs$\"3yw/P*zJ+X \"Fhs7$$\"3E)eZ_w`\"[7Fhs$\"3A)*o\\Yj3l7Fhs7$$\"3!pL7_*pBg7Fhs$\"3!\\F D=&ov^5Fhs7$$\"3a&3x^A?BF\"Fhs$\"3](*pO5BAW\")F37$$\"3=M=9bMS%G\"Fhs$ \"3U.(4*G5>!e&F37$$\"3/$e1^o'['H\"Fhs$\"3#GEf$)Qi$zGF37$$\"3\">Lr]\"*p &38Fhs$\"3q>$pdd&=)*)*Fd]l7$$\"3c!3O]9`1K\"Fhs$!3!\\^+ah$f,FF37$$\"3?H 3+vjtK8Fhs$!3I;-_Ol>iaF37$$\"3%ydl\\g>[M\"Fhs$!39mo))>`J\"=H\"Fhs 7$$\"3ed)4'eD[\"Q\"Fhs$!3'fuIPnF6]\"Fhs7$$\"3eRYK_!zRR\"Fhs$!3k69arNUx ;Fhs7$$\"3#=URgavkS\"Fhs$!3=i(>\"Gz\\;=Fhs7$$\"30/UvR?(*=9Fhs$!3\\b]-( *o*\\\">Fhs7$$\"3G&f6mG?_U\"Fhs$!3)QkP!QaB[>Fhs7$$\"3G')*oM`o9V\"Fhs$! 3u#G5!p\")\\q>Fhs7$$\"3GxjK!y;xV\"Fhs$!3o\"H&>#zP;)>Fhs7$$\"3^oP=F]'RW \"Fhs$!3!*es6C:d\")>Fhs7$$\"3u]&)*3_hkX\"Fhs$!33\"e?M!f\"y%>Fhs7$$\"3) HL8Y,e*o9Fhs$!3AA6MN?()p=Fhs7$$\"3?:\"G$3XX\"[\"Fhs$!3!*\\c2X-T\\Ys%*)R%*=:Fhs$!3klfjCu7k6Fhs7$$\"38Ws=$[S9`\"Fhs$!3_ca wMm8\"4*F37$$\"3OE?!p(p$Ra\"Fhs$!3\\=ghAd#HL'F37$$\"3O3ohqMVc:Fhs$!3!= ')\\s]!pHMF37$$\"3g!fJV'*H*o:Fhs$!3?CyIcw))yWF-7$$\"3'RaK;Q'Q\"e\"Fhs$ \"3gqO['eNY`#F37$$\"3c(\\L*)zUQf\"Fhs$\"3w6,V\"ok%faF37$$\"3$4XMi@*H1; Fhs$\"3-xr$fJb+E)F37$$\"3I/a`Lcv=;Fhs$\"3K\"fkphws3\"Fhs7$$\"3odj$3077 j\"Fhs$\"3z'y?]BMQK\"Fhs7$$\"306t8o%oOk\"Fhs$\"3AVi>4&H.`\"Fhs7$$\"3Uk #Qa)[7c;Fhs$\"366_#3P9@q\"Fhs7$$\"3y<#RFI\"eo;Fhs$\"3SEcbD!\\`$=Fhs7$$ \"3;r,/?x.\"o\"Fhs$\"3\"3LLV9!4F>Fhs7$$\"3_C6MPT\\$p\"Fhs$\"3C]td,xNv> Fhs7$$\"3K,;*fMA(*p\"Fhs$\"3SX+)y$[%G)>Fhs7$$\"37y?ka0&fq\"Fhs$\"3M'RQ C!*y\"z>Fhs7$$\"3paDHj(y@r\"Fhs$\"3S)4]*f*)Rk>Fhs7$$\"3\\JI%>(pS=3'Q>Fhs7$$\"3YQ\\a1)>Lu\"Fhs$\"3@qS2:]fH5iFmW[> =Fhs$\"36xj4evKPIF37$$\"3K^&eDU(HK=Fhs$\"3UOCbm\"4c\"pFd]l7$$\"3o#*3%) y.6X=Fhs$!35086hiO!)GF37$$\"3/MK7NL#z&=Fhs$!384qz7(Q/u&F37$$\"3=vbS\"H O2(=Fhs$!3O)R>MFHHW)F37$$\"3K;zoZ#\\N)=Fhs$!3C\"R78=/C4\"Fhs7$$\"3md-( R?ij*=Fhs$!3@_DGije78Fhs7$$\"3/*f_-;v\"4>Fhs$!3!Q_ul\\v(*\\\"Fhs7$$\"3 #>Fhs$!3Cm#Q!RYv\\;Fhs7$$\"3J\"GFhs$!3%3Fm;Z^#fFhs$!3[JnHZ9,E=Fhs7$$\"3-$zSs]?S&>Fhs$!3v(zEu)4&H%=Fhs7 $$\"3ej>Q&)pUg>Fhs$!33d2`*zW)[=Fhs7$$\"3QMJ_jM$o'>Fhs$!3hn]Z'z8P%=Fhs7 $$\"3%\\Ik;%*RK(>Fhs$!3Td#eHUYw#=Fhs7$$\"37_(*Rf7\"e*>Fhs$!3s#G[MBrno \"Fhs7$$\"3`*>Nrd#Q=?Fhs$!3j]E4XD%eU\"Fhs7$$\"3,BH+O#o'H?Fhs$!3Y)orTJp oD\"Fhs7$$\"3$pkq[*Q&4/#Fhs$!3!Qm@$)=Ym1\"Fhs7$$\"3'3PQPbRA0#Fhs$!3anw LX!y'*e)F37$$\"3M%41E@DN1#Fhs$!312)o]k(>zjF37$$\"3#y\"QZr3\"[2#Fhs$!3# R>'Hsk'z2%F37$$\"3uT:MIl4'3#Fhs$!36qprjt;I'F-7$$\"3q))p2[ym3@Fhs$\"34>#>(ziUFHF37$$\"3j7Z%p]`*>@Fhs$\" 3s!*o$Rne/:&F37$$\"36OC\"e;R78#Fhs$\"3av5?BS>[sF37$$\"3ff,oC[_U@Fhs$\" 3o*4vlU)*H=*F37$$\"3_$)ya$[5Q:#Fhs$\"3yk&R$\\\\3#4\"Fhs7$$\"3GJfJT/Az@ Fhs$\"3'QI50$3f'R\"Fhs7$$\"31zR3*RIY?#Fhs$\"3S`\"Fhs7$$\"30rh:I-F1BFhs$\"3!zL5e? 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 323 115 "Resonance as a limiting situation with respect to \+ the natural frequency and the frequency of the \"forcing\" function" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Consider the general initial value problem" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+omega ^2*x = F[0]*cos*gamma*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF' F)!\"\"F)*&%&omegaGF'F(F)F)**&%\"FG6#\"\"!F)%$cosGF)%&gammaGF)F,F)" } {TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 94 "which involves a system with no damping, \+ but with an external force which varies sinusoidally." }}{PARA 0 "" 0 "" {TEXT -1 53 "We shall investigate how the solution depends on the \+ " }{TEXT 259 17 "natural frequency" }{TEXT -1 1 " " }{XPPEDIT 18 0 "om ega/(2*Pi);" "6#*&%&omegaG\"\"\"*&\"\"#F%%#PiGF%!\"\"" }{TEXT -1 34 " \+ of the system, and the frequency " }{XPPEDIT 18 0 "gamma/(2*Pi);" "6#* &%&gammaG\"\"\"*&\"\"#F%%#PiGF%!\"\"" }{TEXT -1 25 " of the forcing fu nction." }{TEXT 262 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 22 "complementary solution" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "c(t ) = C[1]*cos*omega*t+C[2]*sin*omega*t;" "6#/-%\"cG6#%\"tG,&**&%\"CG6# \"\"\"F-%$cosGF-%&omegaGF-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Assuming that " }{XPPEDIT 18 0 "gamma<>omega" "6#0%&gamma G%&omegaG" }{TEXT -1 59 ", the appropriate form for a trial particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*gamma*t+B*si n*gamma*t;" "6#/%!G,&**%\"AG\"\"\"%$cosGF(%&gammaGF(%\"tGF(F(**%\"BGF( %$sinGF(F*F(F+F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "The n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t );" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "`` = -g amma*A*sin*gamma*t+gamma*B*cos*gamma*t;" "6#/%!G,&*,%&gammaG\"\"\"%\"A GF(%$sinGF(F'F(%\"tGF(!\"\"*,F'F(%\"BGF(%$cosGF(F'F(F+F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) = `p ''`(t);" "6#/*(%\"dG\"\"#% \"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -ga mma^2*A*cos*gamma*t-gamma^2*B*sin*gamma*t;" "6#/%!G,&*,%&gammaG\"\"#% \"AG\"\"\"%$cosGF*F'F*%\"tGF*!\"\"*,F'F(%\"BGF*%$sinGF*F'F*F,F*F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " } {XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF %F'!\"\"" }{TEXT -1 5 " and " }{TEXT 266 1 "x" }{TEXT -1 37 " in the d ifferential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "-gamma^2*A*cos*gamma*t-gamma^2*B*sin*gamma*t+omega^2*A* cos*gamma*t+omega^2*B*sin*gamma*t = F[0]*cos*gamma*t;" "6#/,**,%&gamma G\"\"#%\"AG\"\"\"%$cosGF)F&F)%\"tGF)!\"\"*,F&F'%\"BGF)%$sinGF)F&F)F+F) F,*,%&omegaGF'F(F)F*F)F&F)F+F)F)*,F1F'F.F)F/F)F&F)F+F)F)**&%\"FG6#\"\" !F)F*F)F&F)F+F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "By comparing coefficients of the cosine a nd sine terms we obtain the equations: " }{XPPEDIT 18 0 "(omega^2-gamm a^2)*A = F[0];" "6#/*&,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF(!\"\"F)%\"AGF )&%\"FG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "(omega^2-gamma^2)* B = 0;" "6#/*&,&*$%&omegaG\"\"#\"\"\"*$%&gammaGF(!\"\"F)%\"BGF)\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "A = F[0]/(omega^2-gamma^2);" "6#/%\"AG*&&%\"FG6#\"\"!\"\"\",&*$%&omeg aG\"\"#F**$%&gammaGF.!\"\"F1" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B=0 " "6#/%\"BG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular solution " }{XPPEDIT 18 0 "p(t) = F[0]/(omeg a^2-gamma^2);" "6#/-%\"pG6#%\"tG*&&%\"FG6#\"\"!\"\"\",&*$%&omegaG\"\"# F-*$%&gammaGF1!\"\"F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*gamma*t;" " 6#*(%$cosG\"\"\"%&gammaGF%%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "The general solution " }{XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6 #/-%\"xG6#%\"tG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 44 " for the \+ differential equation is therefore " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(t) = C[1]*cos*omega*t+C[2]*sin*omega*t+F[0]/(omega^2-gamma^2);" " 6#/-%\"xG6#%\"tG,(**&%\"CG6#\"\"\"F-%$cosGF-%&omegaGF-F'F-F-**&F+6#\" \"#F-%$sinGF-F/F-F'F-F-*&&%\"FG6#\"\"!F-,&*$F/F3F-*$%&gammaGF3!\"\"F>F -" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*gamma*t;" "6#*(%$cosG\"\"\"%&ga mmaGF%%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is s traightforward to check that the " }{TEXT 259 19 "particular solution " }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " x(t) = -F[0]/(omega^2-gamma^2);" "6#/-%\"xG6#%\"tG,$*&&%\"FG6#\"\"!\" \"\",&*$%&omegaG\"\"#F.*$%&gammaGF2!\"\"F5F5" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*omega*t+F[0]/(omega^2-gamma^2);" "6#,&*(%$cosG\"\" \"%&omegaGF&%\"tGF&F&*&&%\"FG6#\"\"!F&,&*$F'\"\"#F&*$%&gammaGF0!\"\"F3 F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*gamma*t;" "6#*(%$cosG\"\"\"%&g ammaGF%%\"tGF%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=F[0]/(omega^2-ga mma^2)*(cos*gamma*t-cos*omega*t)" "6#/-%\"xG6#%\"tG*(&%\"FG6#\"\"!\"\" \",&*$%&omegaG\"\"#F-*$%&gammaGF1!\"\"F4,&*(%$cosGF-F3F-F'F-F-*(F7F-F0 F-F'F-F4F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "satisfies \+ the given initial condtions. " }}{PARA 0 "" 0 "" {TEXT -1 42 "This sol ution can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x(t)=2*F[0]/(omega^2-gamma^2)" "6#/-%\"xG6#%\"tG*(\" \"#\"\"\"&%\"FG6#\"\"!F*,&*$%&omegaGF)F**$%&gammaGF)!\"\"F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin((omega-gamma)*t/2)*sin((omega+gamma)*t/2) " "6#*&-%$sinG6#*(,&%&omegaG\"\"\"%&gammaG!\"\"F*%\"tGF*\"\"#F,F*-F%6# *(,&F)F*F+F*F*F-F*F.F,F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 24 "________________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Although the solution is not defined when " } {XPPEDIT 18 0 "gamma = omega;" "6#/%&gammaG%&omegaG" }{TEXT -1 47 ", i t is interesting to observe what happens as " }{XPPEDIT 18 0 "proc (ga mma) options operator, arrow; omega end;" "6#f*6#%&gammaG7\"6$%)operat orG%&arrowG6\"%&omegaGF*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = F[0]/2;" "6#/- %\"xG6#%\"tG*&&%\"FG6#\"\"!\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 271 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin((omega-gamma)*t/2)/``((o mega-gamma)/2);" "6#*&-%$sinG6#*(,&%&omegaG\"\"\"%&gammaG!\"\"F*%\"tGF *\"\"#F,F*-%!G6#*&,&F)F*F+F,F*F.F,F," }{TEXT -1 1 " " }{TEXT 268 1 ". " }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin((omega+gamma)*t/2)/``((omega+gam ma)/2)" "6#*&-%$sinG6#*(,&%&omegaG\"\"\"%&gammaGF*F*%\"tGF*\"\"#!\"\"F *-%!G6#*&,&F)F*F+F*F*F-F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = F [0]*t/2;" "6#/-%\"xG6#%\"tG*(&%\"FG6#\"\"!\"\"\"F'F-\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 270 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin((o mega-gamma)*t/2)/``((omega-gamma)*t/2);" "6#*&-%$sinG6#*(,&%&omegaG\" \"\"%&gammaG!\"\"F*%\"tGF*\"\"#F,F*-%!G6#*(,&F)F*F+F,F*F-F*F.F,F," } {TEXT -1 1 " " }{TEXT 269 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin((o mega+gamma)*t/2)/``((omega+gamma)/2)" "6#*&-%$sinG6#*(,&%&omegaG\"\"\" %&gammaGF*F*%\"tGF*\"\"#!\"\"F*-%!G6#*&,&F)F*F+F*F*F-F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " Using the fact that " }{XPPEDIT 18 0 "Limit(sin*h/h,h=0)=1" "6#/-%&Lim itG6$*(%$sinG\"\"\"%\"hGF)F*!\"\"/F*\"\"!F)" }{TEXT -1 17 ", we see th at as " }{XPPEDIT 18 0 "gamma->omega" "6#f*6#%&gammaG7\"6$%)operatorG% &arrowG6\"%&omegaGF*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin((omega- gamma)*t/2)/``((omega-gamma)*t/2)" "6#*&-%$sinG6#*(,&%&omegaG\"\"\"%&g ammaG!\"\"F*%\"tGF*\"\"#F,F*-%!G6#*(,&F)F*F+F,F*F-F*F.F,F," }{TEXT -1 13 " tends to 1. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "sin((omega+gamma)*t/2)/``((omega+gamma)/2)" "6#*&-%$sinG6#*(,&%&ome gaG\"\"\"%&gammaGF*F*%\"tGF*\"\"#!\"\"F*-%!G6#*&,&F)F*F+F*F*F-F.F." } {TEXT -1 10 " tends to " }{XPPEDIT 18 0 "sin*omega*t/omega;" "6#**%$si nG\"\"\"%&omegaGF%%\"tGF%F&!\"\"" }{TEXT -1 5 ", as " }{XPPEDIT 18 0 " gamma->omega" "6#f*6#%&gammaG7\"6$%)operatorG%&arrowG6\"%&omegaGF*F*F* " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means that " } {XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 10 " tends to " } {XPPEDIT 18 0 "F[0]*t*sin*omega*t/(2*omega);" "6#*.&%\"FG6#\"\"!\"\"\" %\"tGF(%$sinGF(%&omegaGF(F)F(*&\"\"#F(F+F(!\"\"" }{TEXT -1 11 ", that \+ is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit( F[0]/( omega^2-gamma^2)*(cos*gamma*t-cos*omega*t),gamma=omega)=F[0]*t*sin*ome ga*t/(2*omega)" "6#/-%&LimitG6$*(&%\"FG6#\"\"!\"\"\",&*$%&omegaG\"\"#F ,*$%&gammaGF0!\"\"F3,&*(%$cosGF,F2F,%\"tGF,F,*(F6F,F/F,F7F,F3F,/F2F/*. &F)6#F+F,F7F,%$sinGF,F/F,F7F,*&F0F,F/F,F3" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 284 24 "________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "One can check that " }{XPPEDIT 18 0 "x(t)=F[0]*t*sin*omeg a*t/(2*omega)" "6#/-%\"xG6#%\"tG*.&%\"FG6#\"\"!\"\"\"F'F-%$sinGF-%&ome gaGF-F'F-*&\"\"#F-F/F-!\"\"" }{TEXT -1 50 " is the solution of the ini tial value of problem: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+omega^2*x = F[0]*cos*omega*t;" "6#/,&*(% \"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&%&omegaGF'F(F)F)**&%\"FG 6#\"\"!F)%$cosGF)F/F)F,F)" }{TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/% \"tG\"\"!F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "Alternati vely, we can obtain this solution independently. " }}{PARA 0 "" 0 "" {TEXT -1 58 "The complementary solution is the same as before, namely \+ " }{XPPEDIT 18 0 "c(t) = C[1]*cos*omega*t+C[2]*sin*omega*t;" "6#/-%\" cG6#%\"tG,&**&%\"CG6#\"\"\"F-%$cosGF-%&omegaGF-F'F-F-**&F+6#\"\"#F-%$s inGF-F/F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "The a ppropriate form for a trial particular solution is" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" } {XPPEDIT 18 0 "`` = A*t*cos*omega*t+B*t*sin*omega*t;" "6#/%!G,&*,%\"AG \"\"\"%\"tGF(%$cosGF(%&omegaGF(F)F(F(*,%\"BGF(F)F(%$sinGF(F+F(F)F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\" \"%#dtG!\"\"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*omega*t-omega*A *t*sin*omega*t+B*sin*omega*t+omega*B*t*cos*omega*t;" "6#/%!G,***%\"AG \"\"\"%$cosGF(%&omegaGF(%\"tGF(F(*.F*F(F'F(F+F(%$sinGF(F*F(F+F(!\"\"** %\"BGF(F-F(F*F(F+F(F(*.F*F(F0F(F+F(F)F(F*F(F+F(F(" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2) = `p ''`(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\" *&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -2*omega*A*si n*omega*t-omega^2*A*t*cos*omega*t+2*omega*B*cos*omega*t-omega^2*B*t*si n*omega*t;" "6#/%!G,**.\"\"#\"\"\"%&omegaGF(%\"AGF(%$sinGF(F)F(%\"tGF( !\"\"*.F)F'F*F(F,F(%$cosGF(F)F(F,F(F-*.F'F(F)F(%\"BGF(F/F(F)F(F,F(F(*. F)F'F1F(F,F(F+F(F)F(F,F(F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\" \"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 5 " and " }{TEXT 272 1 "x" }{TEXT -1 37 " in the differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-2*omega*A*sin*omega*t-omeg a^2*A*t*cos*omega*t+2*omega*B*cos*omega*t-omega^2*B*t*sin*omega*t;" "6 #,**.\"\"#\"\"\"%&omegaGF&%\"AGF&%$sinGF&F'F&%\"tGF&!\"\"*.F'F%F(F&F*F &%$cosGF&F'F&F*F&F+*.F%F&F'F&%\"BGF&F-F&F'F&F*F&F&*.F'F%F/F&F*F&F)F&F' F&F*F&F+" }{TEXT -1 16 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``+omega^2*A*t*cos*omega*t+omega^2*B*t*sin*om ega*t = F[0]*cos*omega*t;" "6#/,(%!G\"\"\"*.%&omegaG\"\"#%\"AGF&%\"tGF &%$cosGF&F(F&F+F&F&*.F(F)%\"BGF&F+F&%$sinGF&F(F&F+F&F&**&%\"FG6#\"\"!F &F,F&F(F&F+F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-2*omega*A*sin*o mega*t+2*omega*B*cos*omega*t = F[0]*cos*omega*t;" "6#/,&*.\"\"#\"\"\"% &omegaGF'%\"AGF'%$sinGF'F(F'%\"tGF'!\"\"*.F&F'F(F'%\"BGF'%$cosGF'F(F'F +F'F'**&%\"FG6#\"\"!F'F/F'F(F'F+F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 21 "Hence we should take " }{XPPEDIT 18 0 "A=0" "6#/%\"AG\" \"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B = F[0]/(2*omega);" "6#/%\"B G*&&%\"FG6#\"\"!\"\"\"*&\"\"#F*%&omegaGF*!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular solution " }{XPPEDIT 18 0 "p(t) = F[0]*t*sin*omega*t/(2*omega);" "6#/-%\"pG6#%\"t G*.&%\"FG6#\"\"!\"\"\"F'F-%$sinGF-%&omegaGF-F'F-*&\"\"#F-F/F-!\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "ge neral solution" }{TEXT -1 10 " given by " }{XPPEDIT 18 0 "x(t)=c(t)+p( t)" "6#/-%\"xG6#%\"tG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 3 " is " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(t) = C[1]*cos*omega*t+C[2]*sin* omega*t+F[0]*t*sin*omega*t/(2*omega);" "6#/-%\"xG6#%\"tG,(**&%\"CG6#\" \"\"F-%$cosGF-%&omegaGF-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-*.&%\"FG 6#\"\"!F-F'F-F4F-F/F-F'F-*&F3F-F/F-!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straightforward to check that the " } {TEXT 259 19 "particular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = F[0]*t*sin*omega*t/(2*omega); " "6#/-%\"xG6#%\"tG*.&%\"FG6#\"\"!\"\"\"F'F-%$sinGF-%&omegaGF-F'F-*&\" \"#F-F/F-!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 273 10 "__________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "obtained by taking " }{XPPEDIT 18 0 "C[1]=C[2]" "6#/&%\"CG6#\"\"\" &F%6#\"\"#" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 41 ", satisf ies the given initial condtions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "Calculation with Maple " }}{PARA 0 "" 0 "" {TEXT -1 40 "First we solve the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+omega^2*x = F[0]*cos*gamma*t, x(0) = 0;" "6$/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&%&om egaGF'F(F)F)**&%\"FG6#\"\"!F)%$cosGF)%&gammaGF)F,F)/-F(6#F4F4" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%# dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := diff( x(t),t$2)+omega^2*x(t)=F[0]*cos(gamma*t);\nic := x(0)=0,D(x)(0)=0;\nds olve(\{de,ic\},x(t));\nsol := simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*& )%&omegaGF1F2F*F2F2*&&%\"FG6#\"\"!F2-%$cosG6#*&%&gammaGF2F-F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F) F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*(-%$cosG6#*&%&o megaG\"\"\"F'F/F/&%\"FG6#\"\"!F/,&*$)F.\"\"#F/!\"\"*$)%&gammaGF7F/F/F8 F/*(F0F/-F+6#*&F;F/F'F/F/F4F8F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ solG/-%\"xG6#%\"tG*(&%\"FG6#\"\"!\"\"\",&-%$cosG6#*&%&omegaGF/F)F/F/-F 26#*&%&gammaGF/F)F/!\"\"F/,&*$)F5\"\"#F/F:*$)F9F>F/F/F:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We can also check \+ that the solution can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=2*F[0]/(omega^2-gamma^2)" "6#/-%\" xG6#%\"tG*(\"\"#\"\"\"&%\"FG6#\"\"!F*,&*$%&omegaGF)F**$%&gammaGF)!\"\" F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin((omega-gamma)*t/2)*sin((omega+ gamma)*t/2)" "6#*&-%$sinG6#*(,&%&omegaG\"\"\"%&gammaG!\"\"F*%\"tGF*\" \"#F,F*-F%6#*(,&F)F*F+F*F*F-F*F.F,F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "2*F[0]/(o mega^2-gamma^2)*sin((omega-gamma)*t/2)*sin((omega+gamma)*t/2);\ncombin e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\"&%\"FG6#\"\"!F &,&*$)%&omegaGF%F&F&*$)%&gammaGF%F&!\"\"F2-%$sinG6#,$*(F%F2,&F.F&F1F2F &%\"tGF&F&F&-F46#,$*(F%F2,&F.F&F1F&F&F9F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&-%$cosG6#*&%&omegaG\"\"\"%\"tGF+F+&%\"FG6#\"\"!F+ F+*&F-F+-F'6#*&%&gammaGF+F,F+F+!\"\"F+,&*$)F*\"\"#F+F6*$)F5F:F+F+F6" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We can \+ check that the solution tends to " }{XPPEDIT 18 0 "F[0]*t*sin(omega*t) /(2*omega);" "6#**&%\"FG6#\"\"!\"\"\"%\"tGF(-%$sinG6#*&%&omegaGF(F)F(F (*&\"\"#F(F.F(!\"\"" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "gamma->omega" "6#f*6#%&gammaG7\"6$%)operatorG%&arrowG6\"%&omegaGF*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Limit(rhs(sol),gamma=omega);\nvalue(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%&LimitG6$*(&%\"FG6#\"\"!\"\"\",&-%$cosG6#*&%&o megaGF+%\"tGF+F+-F.6#*&%&gammaGF+F2F+!\"\"F+,&*$)F1\"\"#F+F7*$)F6F;F+F +F7/F6F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&**-%$sin G6#*&%&omegaGF&%\"tGF&F&F.F&F-!\"\"&%\"FG6#\"\"!F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We obtain the same r esult if we put " }{XPPEDIT 18 0 "gamma = omega;" "6#/%&gammaG%&omegaG " }{TEXT -1 72 " in the differential equation, and then solve the init ial value problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 111 "de := diff(x(t),t$2)+omega^2*x(t)=F[0]*cos( omega*t);\nic := x(0)=0,D(x)(0)=0;\ndsolve(\{de,ic\},x(t));\nsimplify( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"t G-%\"$G6$F-\"\"#\"\"\"*&)%&omegaGF1F2F*F2F2*&-%$cosG6#*&F5F2F-F2F2&%\" FG6#\"\"!F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F */--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&* &#\"\"\"\"\"#F+*(-%$cosG6#*&%&omegaGF+F'F+F+&%\"FG6#\"\"!F+F2!\"#F+!\" \"*&#F+F,F+*(F3F+,&F.F+*(-%$sinGF0F+F2F+F'F+F+F+F2F7F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*&#\"\"\"\"\"#F+**-%$sinG6#*&%&o megaGF+F'F+F+F'F+F2!\"\"&%\"FG6#\"\"!F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Examples to compare of the phenomena of resonance and bea ts " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Consider the two initial value problems: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+10000*x = 800*cos* 98*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"&++\"F )F(F)F)**\"$+)F)%$cosGF)\"#)*F)F,F)" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "x(0) = 0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ev al(dx/dt,t = 0) = 0" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F ." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+10000*x = 800*cos*10 0*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"&++\"F) F(F)F)**\"$+)F)%$cosGF)\"$+\"F)F,F)" }{TEXT -1 15 ", x(0) = 0, " } {XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG !\"\"/%\"tG\"\"!F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "in which the only difference is in the fr equency of the \"forcing\" term. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "In each case the " }{TEXT 259 22 "complem entary solution" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "c(t) = C[1]*cos*10 0*t+C[2]*sin*100*t;" "6#/-%\"cG6#%\"tG,&**&%\"CG6#\"\"\"F-%$cosGF-\"$+ \"F-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "In the first di fferential equation with the forcing function " }{XPPEDIT 18 0 "800*co s*98*t;" "6#**\"$+)\"\"\"%$cosGF%\"#)*F%%\"tGF%" }{TEXT -1 58 " the ap propriate form for a trial particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" } {XPPEDIT 18 0 "``=A*cos*98*t+B*sin*98*t" "6#/%!G,&**%\"AG\"\"\"%$cosGF (\"#)*F(%\"tGF(F(**%\"BGF(%$sinGF(F*F(F+F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$p~'G6 #%\"tG" }{XPPEDIT 18 0 "``=-98*A*sin*98*t+98*B*cos*98*t" "6#/%!G,&*,\" #)*\"\"\"%\"AGF(%$sinGF(F'F(%\"tGF(!\"\"*,F'F(%\"BGF(%$cosGF(F'F(F+F(F (" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) = `p ''`(t);" "6#/ *(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "``=-9604*A*cos*98*t-9604*B*sin*98*t" "6#/%!G,&*,\"%/'*\"\"\"%\"A GF(%$cosGF(\"#)*F(%\"tGF(!\"\"*,F'F(%\"BGF(%$sinGF(F+F(F,F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\" " }{TEXT -1 5 " and " }{TEXT 285 1 "x" }{TEXT -1 43 " in the first dif ferential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "-9604*A*cos*98*t-9604*B*sin*98*t+10000*A*cos*98*t+10000 *B*sin*98*t = 800*cos*98*t;" "6#/,**,\"%/'*\"\"\"%\"AGF'%$cosGF'\"#)*F '%\"tGF'!\"\"*,F&F'%\"BGF'%$sinGF'F*F'F+F'F,*,\"&++\"F'F(F'F)F'F*F'F+F 'F'*,F1F'F.F'F/F'F*F'F+F'F'**\"$+)F'F)F'F*F'F+F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "By compar ing coefficients of the cosine and sine terms we obtain the equations: " }{XPPEDIT 18 0 "396*A=800" "6#/*&\"$'R\"\"\"%\"AGF&\"$+)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "396*B=0" "6#/*&\"$'R\"\"\"%\"BGF&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 " A=200/99" "6#/%\"AG*&\"$+#\"\"\"\"#**!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "B=0" "6#/%\"BG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 39 "Thus we obtain the particular solution " }{XPPEDIT 18 0 "p(t) = 200/99;" "6#/-%\"pG6#%\"tG*&\"$+#\"\"\"\"#**!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*98*t" "6#*(%$cosG\"\"\"\"#)*F%%\"tGF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "The general solution fo r the first differential equation given by " }{XPPEDIT 18 0 "x(t)=c(t) +p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 14 " is therefore " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x(t) = C[1]*cos*100*t+C[2]*sin*100*t+200/99;" "6#/-%\"x G6#%\"tG,(**&%\"CG6#\"\"\"F-%$cosGF-\"$+\"F-F'F-F-**&F+6#\"\"#F-%$sinG F-F/F-F'F-F-*&\"$+#F-\"#**!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos *98*t" "6#*(%$cosG\"\"\"\"#)*F%%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straightforward to check that the " }{TEXT 259 19 "particular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = -200/99;" "6#/-%\"xG6#%\"tG,$*& \"$+#\"\"\"\"#**!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*100*t+200 /99" "6#,&*(%$cosG\"\"\"\"$+\"F&%\"tGF&F&*&\"$+#F&\"#**!\"\"F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*98*t" "6#*(%$cosG\"\"\"\"#)*F%%\"tG F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "satisfies the give n initial condtions. " }}{PARA 0 "" 0 "" {TEXT -1 41 "This solution ca n be written in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x(t)=400/99" "6#/-%\"xG6#%\"tG*&\"$+%\"\"\"\"#**!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*t*sin*99*t" "6#*,%$sinG\"\"\"%\"tG F%F$F%\"#**F%F&F%" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 287 12 "____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Turning attention to the \+ second differential equation with the forcing function " }{XPPEDIT 18 0 "800*cos*100*t;" "6#**\"$+)\"\"\"%$cosGF%\"$+\"F%%\"tGF%" }{TEXT -1 156 " we notice that it duplicates one of the function which appears i n the complementary solution. Hence the appropriate form for a trial p articular solution is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*t*cos*1 00*t+B*t*sin*100*t;" "6#/%!G,&*,%\"AG\"\"\"%\"tGF(%$cosGF(\"$+\"F(F)F( F(*,%\"BGF(F)F(%$sinGF(F+F(F)F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\"\"-%$p~'G6#%\"tG" } {XPPEDIT 18 0 "``=A*cos*100*t-100*A*t*sin*100*t+B*sin*100*t+100*B*t*co s*100*t" "6#/%!G,***%\"AG\"\"\"%$cosGF(\"$+\"F(%\"tGF(F(*.F*F(F'F(F+F( %$sinGF(F*F(F+F(!\"\"**%\"BGF(F-F(F*F(F+F(F(*.F*F(F0F(F+F(F)F(F*F(F+F( F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) = `p ''`(t);" "6#/ *(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "``=-200*A*sin*100*t-10000*A*t*cos*100*t+200*B*cos*100*t-10000*B* t*sin*100*t" "6#/%!G,**,\"$+#\"\"\"%\"AGF(%$sinGF(\"$+\"F(%\"tGF(!\"\" *.\"&++\"F(F)F(F,F(%$cosGF(F+F(F,F(F-*,F'F(%\"BGF(F0F(F+F(F,F(F(*.F/F( F2F(F,F(F*F(F+F(F,F(F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"x G\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 5 " and " }{TEXT 286 1 "x" } {TEXT -1 44 " in the second differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-200*A*sin*100*t-10000*A*t* cos*100*t+200*B*cos*100*t-10000*B*t*sin*100*t" "6#,**,\"$+#\"\"\"%\"AG F&%$sinGF&\"$+\"F&%\"tGF&!\"\"*.\"&++\"F&F'F&F*F&%$cosGF&F)F&F*F&F+*,F %F&%\"BGF&F.F&F)F&F*F&F&*.F-F&F0F&F*F&F(F&F)F&F*F&F+" }{TEXT -1 16 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``+ 10000*A*t*cos*100*t+10000*B*t*sin*100*t=800*cos*100*t" "6#/,(%!G\"\"\" *.\"&++\"F&%\"AGF&%\"tGF&%$cosGF&\"$+\"F&F*F&F&*.F(F&%\"BGF&F*F&%$sinG F&F,F&F*F&F&**\"$+)F&F+F&F,F&F*F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-200*A*sin*100*t+200*B*cos*100*t = 800*cos*100*t" "6#/,&*,\"$+# \"\"\"%\"AGF'%$sinGF'\"$+\"F'%\"tGF'!\"\"*,F&F'%\"BGF'%$cosGF'F*F'F+F' F'**\"$+)F'F/F'F*F'F+F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Hence we should take " }{XPPEDIT 18 0 "A=0" "6#/%\"AG\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B=4" "6#/%\"BG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular soluti on " }{XPPEDIT 18 0 "p(t)=4*t*sin*100*t" "6#/-%\"pG6#%\"tG*,\"\"%\"\" \"F'F*%$sinGF*\"$+\"F*F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 67 "The general solution for the second differential equation given by " }{XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6#F'\" \"\"-%\"pG6#F'F," }{TEXT -1 14 " is therefore " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) \+ = C[1]*cos*100*t+C[2]*sin*100*t+4*t*sin*100*t;" "6#/-%\"xG6#%\"tG,(**& %\"CG6#\"\"\"F-%$cosGF-\"$+\"F-F'F-F-**&F+6#\"\"#F-%$sinGF-F/F-F'F-F-* ,\"\"%F-F'F-F4F-F/F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "It is straightforward to check that the " }{TEXT 259 19 "partic ular solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x(t)=4*t*sin*100*t" "6#/-%\"xG6#%\"tG*,\"\"%\"\"\"F'F*% $sinGF*\"$+\"F*F'F*" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 288 10 "__________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "obtained by taking " }{XPPEDIT 18 0 "C[1]=C[2]" "6#/&%\"CG6#\" \"\"&F%6#\"\"#" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 41 ", sa tisfies the given initial condtions. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 91 "It is instructive to compare the solu tions for the two initial value problems graphically. " }}{PARA 0 "" 0 "" {TEXT -1 18 "Over the interval " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!% \"tG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 56 ", there is l ittle difference between the two solutions " }{XPPEDIT 18 0 "g(t) = 4 *t*sin*100*t;" "6#/-%\"gG6#%\"tG*,\"\"%\"\"\"F'F*%$sinGF*\"$+\"F*F'F* " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "h(t) = 400/99;" "6#/-%\"hG6#% \"tG*&\"$+%\"\"\"\"#**!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*t*sin *99*t" "6#*,%$sinG\"\"\"%\"tGF%F$F%\"#**F%F&F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "g := t -> 4*t*sin(100*t):\n'g(t)'=g(t);\nh := t -> 400/99*sin(t)* sin(99*t):\n'h(t)'=h(t);\nplot([g(t),h(t)],t=0..1,color=[red,COLOR(RGB ,.5,0,1)],labels=[`t`,`x(t)`],\n legend=[`resonant solution`,`near re sonant solution`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG, $*(\"\"%\"\"\"F'F+-%$sinG6#,$*&\"$+\"F+F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG,$*&#\"$+%\"#**\"\"\"*&-%$sinGF&F--F06#,$ *&F,F-F'F-F-F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 564 301 301 {PLOTDATA 2 "6&-%'CURVESG6%7gil7$$\"\"!F)F(7$$\"3ILLL3x&)*3\"!#>$\"3gE Ux%y,\\'QF-7$$\"3emmm;arz@F-$\"3U:(o_Q#z^rF-7$$\"3.++D\"y%*z7$F-$\"3%f MW\"=0J,;F@7$$\"3xm;H2Q\\4YF-$!3:n%[^gXS$=F@7$$\"33L3-QO5w[F-$!3XwI=T9O C>F@7$$\"33++voMrU^F-$!3)Rt%e,=ap=F@7$$\"3QL$3-8Lfn&F-$!3!z54q#4X? \"F@7$$\"3emmmm6m#G(F-$\"3]:/dDRV]CF@7$$\"3gnmTg#Q5b(F-$\"3)Gw%3?K(G)G F@7$$\"3Anm;a`T>yF-$\"3GK()*ov(*e7$F@7$$\"3'om;zW#z(3)F-$\"3'e%*)e;74Z JF@7$$\"3[mmmT&phN)F-$\"3)\\@\"Q2%y(HHF@7$$\"36++v=ddC%*F-$\"33+^i[3=C w!#A7$$\"3CLLe*=)H\\5F@$!3%3(RT%Ga\"yOF@7$$\"3)***\\(=#>1u5F@$!3?1N]%) )pt:%F@7$$\"3gmm;ac#))4\"F@$!3W7=FM\\=&R%F@7$$\"3#**\\7._276\"F@$!3l') GVN0q9WF@7$$\"3BL$ekQ*eB6F@$!3g@4?B&)>lVF@7$$\"3amTg_7(f8\"F@$!3')*R& \\-r%fC%F@7$$\"3')***\\(=JN[6F@$!3]:iEGqKdSF@7$$\"3?LLL$e!)y>\"F@$!3k* HY%=E6cEF@7$$\"3gmm\"z/3uC\"F@$!3MSQz3SS)f%F-7$$\"3#)***\\(o3p)H\"F@$ \"3;x0\\L5x?@F@7$$\"33LLe*ot*\\8F@$\"3\")e@0Pf]RVF@7$$\"3q*******4:cP \"F@$\"3[gZsMU'y5&F@7$$\"3KmmT5lD,9F@$\"3g6A#3cr:c&F@7$$\"3x**\\i:s299 F@$\"3QqzQ.@FccF@7$$\"3ALL$3#z*oU\"F@$\"3etj\"4B!3ecF@7$$\"3nm;/E'=(R9 F@$\"3k0y0.+HlbF@7$$\"3%)***\\7LRDX\"F@$\"3#3:)fflxx`F@7$$\"3emm\"z4wb ]\"F@$\"3oZX*[Cm^l$F@7$$\"3/LLekGhe:F@$\"3/+$Hg\"H%pd(F-7$$\"3x***\\7j \\;h\"F@$!3-ak.\\%)*4c#F@7$$\"3]mm\"zR'ok;F@$!3)QNdF^.IP&F@7$$\"3-L$ek t>6p\"F@$!3:;)G33`EJ'F@7$$\"3#)*****\\2`vr\"F@$!3J.]bN'RO$oF@7$$\"33L3 FW(p2t\"F@$!3j.k\\*[!=?pF@7$$\"3hm;a8k)Ru\"F@$!3#pcJ_47c)oF@7$$\"3))* \\7G3.sv\"F@$!3QnmKwaqGnF@7$$\"39LL3_(>/x\"F@$!3iIk[pR(f'F-7$$\"3KL3x\" )H`I>F@$\"3!>p(eqI&*)R$F@7$$\"3gm;HdG\"\\)>F@$\"3ePy)=X_\"zmF@7$$\"37$ 3_]z-@,#F@$\"3M%Gb1EY0p(F@7$$\"3!**\\7Gt#HR?F@$\"3ir\"HB^.T:)F@7$$\"3I 3Fp,x)G0#F@$\"3g+flj>Cj\")F@7$$\"3o;HdqE[m?F@$\"3D5yoh/9?!)F@7$$\"31DJ XRw2!3#F@$\"3?8dJ)[oas(F@7$$\"3WLLL3En$4#F@$\"3'yP!pF%eFG(F@7$$\"3xm;H K/dT@F@$\"3OO>!G>N=m%F@7$$\"33++Dc#o%*=#F@$\"3CB2b/SFN%)F-7$$\"3RL$3-3 mtB#F@$!3?PG/c#3/M$F@7$$\"3qmm;/RE&G#F@$!3PYqm6COOpF@7$$\"3ZL3x1ZA7BF@ $!3!f_w!)eY\"p$)F@7$$\"3)***\\P4b=RBF@$!3p?)Hp@?JYhO#F@$!37j@byev<%*F@7$$\"3')\\7G8nizBF @$!3u))y-[WQe#*F@7$$\"3CLLe9r5$R#F@$!3.e0?[zlF*)F@7$$\"3am;z>(GqW#F@$! 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aat$\"3MBAfJ+)[K\"Ffal7$Ffat$\"3/\"yk`*eT1!*F@7$F[bt$\"3g7&4I]#f$f%F@7 $F`bt$\"3u84$R/3r(*)F:7$$\"3KC1R+2i])*F@$!3IJda6%=QU%F@7$Febt$!3'p`?_j Ac'))F@7$$\"3%[(o/t0yx)*F@$!3D<.2,R_:8Ffal7$Fjbt$!3*3LT'or[@h4'=(R$Ffal-%&COLORG6&F_et$\"\"&!\"\"F(Fhdt-Fdet6#%7near~re sonant~solutionG-%+AXESLABELSG6$%\"tG%%x(t)G-%%VIEWG6$;F(Fhdt%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "resonant s olution" "near resonant solution" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "However, if we plot the graphs of " }{XPPEDIT 18 0 "g(t); " "6#-%\"gG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h(t);" "6#-%\" hG6#%\"tG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0<=t" "6 #1\"\"!%\"tG" }{XPPEDIT 18 0 "`` <= Pi;" "6#1%!G%#PiG" }{TEXT -1 40 " \+ we see that the two solutions diverge. " }}{PARA 0 "" 0 "" {TEXT -1 19 "The first solution " }{XPPEDIT 18 0 "h(t) = 400/99;" "6#/-%\"hG6#% \"tG*&\"$+%\"\"\"\"#**!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*t*sin *99*t" "6#*,%$sinG\"\"\"%\"tGF%F$F%\"#**F%F&F%" }{TEXT -1 27 " exhibit s the phonomena of " }{TEXT 259 5 "beats" }{TEXT -1 27 " while the sec ond solution " }{XPPEDIT 18 0 "g(t) = 4*t*sin*100*t;" "6#/-%\"gG6#%\"t G*,\"\"%\"\"\"F'F*%$sinGF*\"$+\"F*F'F*" }{TEXT -1 25 " exhibits the ph omena of " }{TEXT 259 9 "resonance" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "g := t -> 4*t*sin(100*t):\n'g(t)'=g(t);\nh := t -> 400/99*sin(t)*sin(99*t):\n'h (t)'=h(t);\nplot([g(t),h(t)],t=0..Pi,color=[red,COLOR(RGB,.5,0,1)],num points=150,labels=[`t`,`x(t)`],\n legend=[`resonant solution`,`near r esonant solution`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"tG ,$*(\"\"%\"\"\"F'F+-%$sinG6#,$*&\"$+\"F+F'F+F+F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"hG6#%\"tG,$*&#\"$+%\"#**\"\"\"*&-%$sinGF&F--F06# ,$*&F,F-F'F-F-F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 637 361 361 {PLOTDATA 2 "6&-%'CURVESG6%7gjo7$$\"\"!F)F(7$$\"36fu(yy(*H5\"!#>$\"3Y3 M0p1)z$RF-7$$\"3B=\\vvb*f?#F-$\"3KtN%[N&*G5(F-7$$\"3;s**G`!3d;$F-$!3SI OuHmR`I!#?7$$\"33E]#3`?a7%F-$!3#3CQQ#)oRP\"!#=7$$\"3%>W;z0p]m%F-$!3S*e 9<$)QR'=F@7$$\"3\"y&y+&eE!z6F@7$$\"3a*o!>RY,%G'F-$\"3#eaX/c8Y3#!#@7$$\"3T<$=qISs#oF-$\"3&= kho\\QNT\"F@7$$\"3IXf%[(fYqtF-$\"3v^q=^RA5EF@7$$\"3a$)fJF@7$$\"31,7]5t\"pX)F-$\"3?Aba'p)H'y#F@7$$\"3Y%42E$\\?Q&*F-$!3&e W(H!G&G=VF-7$$\"3#*)HraD\\>1\"F@$!3al#Rui6F@$!3OM:zt0;mPF@7$$\"3'pk#\\hIK77F@$!3w'Hi* *=r#z?F@7$$\"31j(*\\jwWi7F@$\"3F(\\T@`\"eKHF-7$$\"3:uya&y[VJ\"F@$\"3A^ `=m\"*[oGF@7$$\"3'\\)ff2*\\iO\"F@$\"3ul0]\\U\"3'[F@7$$\"3y&4W'H5:=9F@$ \"3!p1.w\"z-ncF@7$$\"3g1Ap^@0q9F@$\"3#[\\IW\"3`r\\F@7$$\"3kjr-\"e/ud\" F@$!3GkJ!y=Al;%F-7$$\"3%47i.,dZo\"F@$!3n'ey3q*>AhF@7$$\"3Lp_aZ!4:r\"F@ $!3[q$[NKYXv'F@7$$\"3+=%GZ3h#Q78]w\"F@$!3 eFQ5:?xylF@7$$\"3y9Z4f^w\"z\"F@$!3[Y)*45yS`dF@7$$\"3c65YL#p_%=F@$!3S]P p\")**)H&GF@7$$\"3i3t#yIt()*=F@$\"3s?P+/?8Y5F@7$$\"3Qb(*zC)3Q&>F@$\"3% pOQT9Oe'\\F@7$$\"3%=?spO(F@7$$\"3w%4w%R#*F@7$$\"3IIeP]$)QnBF@$!3uKZi'Q(G5%*F@7$$\"3mBO46Un%R#F@$!3 -#p,Nb]#y))F@7$$\"3w;9\"=2g>U#F@$!3w!*Rmd9AnwF@7$$\"3C.qC$zJlZ#F@$!3Ya P=\"z)ReNF@7$$\"3)**e#o9N5JDF@$\"3#)o-6!*Gd&z\"F@7$$\"3!R`\\GJ**ee#F@$ \"35/Tc$*H*)ooF@7$$\"3!yZ;56&pSEF@$\"34LRWB1;55!#<7$$\"3/]**45I4oEF@$ \"3!yN*y%[kp1\"F^z7$$\"3EAM=44\\&p#F@$\"35P@'*45JW5F^z7$$\"3]%*oE3)))G s#F@$\"3k[gy,%4GU*F@7$$\"3=m.N2nG]FF@$\"3;lW&GNi)pwF@7$$\"3-`qC<&)*e&G F@$!3%3DNoc0!3KF@7$$\"3')RP9F.^hHF@$!3[#47$e>S`6F^z7$$\"3#p8A$ofTdIF@$ !3xp*eqQV67*F@7$$\"3UL0]4;K`JF@$\"3%**H[O9x7:U.@$F@$\" 3k3^)Het\"\\\")F@7$$\"3_jQ0$pitE$F@$\"3Jrg&eP2MC\"F^z7$$\"3w&>UR'H(eH$ F@$\"32v2bwE$yJ\"F^z7$$\"3+G0$[B$QCLF@$\"3Knz&*>I/'G\"F^z7$$\"3Cg)=d]$ *GN$F@$\"3)>gOH9%z[6F^z7$$\"3/$>2mx.9Q$F@$\"3g2\"f;sw[:*F@7$$\"3y>/Yd$F@$!3i6t!eE?nK\"F^z7$$\"3c$[)3xq*p o$F@$!3)Rk>g-Xu3\"F^z7$$\"3;@LWk[P*z$F@$\"30y:z^rB8WF@7$$\"3?r%*=c:6\\ QF@$\"3GedCs2'e4\"F^z7$$\"3C@c$zC[))*QF@$\"37\"REpu(=)\\\"F^z7$$\"3`'p 3Qf;P#RF@$\"3a;zmDako:F^z7$$\"3#=x\"oR\\e[RF@$\"3=^fkj=vU:F^z7$$\"3cY[ b&G`M(RF@$\"3DTRk'o!*3U\"F^z7$$\"3'=#zUJ;K)*RF@$\"3l'3[\"=XT47F^z7$$\" 3dS@!))pfu5%F@$!37$G:(>V(y!QF@7$$\"3Gfj#=F^z7$$\"3m9@i145(e%F@$\"38+3H`Z* Gu\"F^z7$$\"3l$3[.o4Uh%F@$\"3'3$\\)*)\\wY`\"F^z7$$\"3>`S2a%=8k%F@$\"3- l4(o*)R6@\"F^z7$$\"3$G'*o4s&*3u%F@$!3LBO[7$zHL&F@7$$\"3[sQ'y)HZS[F@$!3 owzrdcOb=F^z7$$\"3yC:jA0)y%\\F@$!3/X[:A3E,9F^z7$$\"3_w\"*Rd!)Gb]F@$\"3 OOS$ezZ=t&F@7$$\"3z9@ay626^F@$\"38xC=%e%[H:F^z7$$\"3;a]o*Hao;&F@$\"3ou >AIctP?F^z7$$\"3a$*z#3UPEA&F@$\"32#[\"))o+6K>F^z7$$\"3!=$4(>a?%y_F@$\" 3FO?%>Q0n_!RtF@7$$\"3wL)Gd-eEZ&F@ $!3U2JB([4.7#F^z7$$\"3QwHhF*[vd&F@$!3S$ePgQu\"e:F^z7$$\"37?r\\H)RCo&F@ $\"3'[2.+tv\")='F@7$$\"3^Z8676iOdF@$\"3%Rx&e5=(Qn\"F^z7$$\"3!\\dDZR-3z &F@$\"3u()>:odtkAF^z7$$\"3IK\"z`rZV!eF@$\"3+*)p%HVR]J#F^z7$$\"3f)oKg.$ *y\"eF@$\"3Vc(GptVIK#F^z7$$\"3)[C'oc$Q9$eF@$\"3-Knk\"*RS)G#F^z7$$\"3H- )Rtn$)\\%eF@$\"3(Qy?5Ke:@#F^z7$$\"3)f\"pk=V2seF@$\"3x#*H[V[(o$>F^z7$$ \"3mHS&*f\\;**eF@$\"3[Bo:T*Hw^\"F^z7$$\"3#zGjXxq@&fF@$\"3%QXf;]US*RF@7 $$\"3EZD<*ew^+'F@$!3K,^O;2r&\\)F@7$$\"3]0=y.C=egF@$!3E83tU&Ra)=F^z7$$ \"3uj5R=#)=6hF@$!3!\\A\\C=xX]E'F@$!3A,kIvB\"4_%F@7$$ \"3o@<'HiJjJ'F@$\"32\"z?Dj;?A)F@7$$\"3%**Rdm\"eFtjF@$\"3/*QUV%oP)*>F^z 7$$\"3LzIN5+AIkF@$\"3[8\"R\"\\E7fDF^z7$$\"3se([S?kr['F@$\"3=L6*fg#o9BF ^z7$$\"3)pVWxR3Ta'F@$\"3(=WPcNO'G8F^z7$$\"31M`Z)Gv_f'F@$\"3_'flBk4'eaF -7$$\"3;Ji?z@WYmF@$!3K[pF8T[`7F^z7$$\"3MHr$*p!4wp'F@$!3iw\"*[s\"e\"eAF ^z7$$\"3WE!o1'fx[nF@$!3Ewd'*\\a?&p#F^z7$$\"3f5u(y`2M!oF@$!3IAt1T9O,CF^ z7$$\"3k$z'3:\"R!eoF@$!3aL!>@KrxR\"F^z7$$\"3qwhH#pqE\"pF@$\"3ExF%*p MDF^z7$$\"33V!3Ze>`;(F@$\"3C&3%eNRXE;F^z7$$\"3jaQ*=*3W>sF@$\"3s\"[K\\F wcz\"F@7$$\"3;m'z!*>iNF(F@$!3Yiu+gr!4M\"F^z7$$\"3gwaE1NoFtF@$!3s(eoU4/ z\\#F^z7$$\"39)G^M\"[!=Q(F@$!3_us1^?f_HF^z7$$\"3qS5uX(GFV(F@$!3\"Ghb6k K$4EF^z7$$\"3E$zI!yEl$[(F@$!3I+!=!\\mQ%f\"F^z7$$\"3#ea?.hwX`(F@$!3ypi, >xD!e\"F@7$$\"3Q)H5Ea+be(F@$\"3n$zVymq#Q8F^z7$$\"3;8s\"\\sI(QwF@$\"3]= (zlVLHb#F^z7$$\"31HTA24'>p(F@$\"3UAh8L#HI2$F^z7$$\"3'\\/J&*3\">XxF@$\" 3f!GGr\"3$Qu#F^z7$$\"3ufz$=F@%)z(F@$\"3!Hq&QH.LX;F^z7$$\"31vWj/NY]yF@$ \"3EFp>c*HX5\"F@7$$\"3F*)4VPd]-zF@$!3'3@ZZpfVZ\"F^z7$$\"3[.vAqzaazF@$! 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35t[_GOhhpF@7$$\"3aq^k\"o=k'HF^z$!3])\\(Q8#=u-(F@7$Fhfal$!31QP_,$))R(p F@7$F]gal$!3L)H+UR$*>_'F@7$Fbgal$!33[Y)p$))e\\cF@7$Fjial$\"3_M(>IX/e# \\F-7$Fb\\bl$\"3)*)4J=WzIQ&F@7$Ff]bl$\"3GY_(yA%=lWF@7$F^`bl$!3)Rn>&y>W k`F-7$Fbabl$!39Fe0KU68FF@7$Ffbbl$!3qciSx1X9TF@7$F[cbl$!3HMX68%QRV%F@7$ F`cbl$!3Ys+!fi8I[%F@7$Fjcbl$!3%pe;z2w+F%F@7$Fddbl$!3O6'4v7X)=QF@7$F\\g bl$\"3gx\"=9h&\\pSF:7$Fdibl$\"3wUoRV&fv/$F@7$Fiibl$\"3-]%eXWKt@$F@7$F^ jbl$\"37!=&pJv.!=$F@7$Fcjbl$\"35,\"HYc$>]HF@7$Fhjbl$\"3;R'e@@\")Rb#F@7 $F\\\\cl$\"3VSLIFI*3T\"F@7$F`]cl$\"3iP\\yj:g9$*F:7$Fd^cl$!30<3tOAs<6F@ 7$Fh_cl$!37F?r\"HC6$=F@7$F]`cl$!3]z*eN3,m&>F@7$Fb`cl$!3*GE:K,*4I>F@7$F g`cl$!39]4.(oo\"o5,tF-7$F`gcl$\"3*[3F_R\"eNRF-7$Fhicl$\"3$yLIe#3KVR!#K- %&COLORG6&F`jcl$\"\"&!\"\"F($\"\"\"F)-Fejcl6#%7near~resonant~solutionG -%+AXESLABELSG6$%\"tG%%x(t)G-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "resonant solution " "near resonant solution" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 267 "These examples illustrate how the phonomena of resonance occurs as th e limiting situation as the frequency of the forcing function approach es the natural frequency associated with the corresponding homogeneous differential equation (simple harmonic motion equation). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The preceding cal culations can be performed with Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "de := diff(x(t),t$2)+10 000*x(t)=800*cos(98*t);\nic := x(0)=0,D(x)(0)=0;\ndsolve(\{de,ic\},x(t ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"t G-%\"$G6$F-\"\"#\"\"\"*&\"&++\"F2F*F2F2,$*&\"$+)F2-%$cosG6#,$*&\"#)*F2 F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F* /--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*& #\"$+#\"#**\"\"\"-%$cosG6#,$*&\"$+\"F-F'F-F-F-!\"\"*&#F+F,F--F/6#,$*& \"#)*F-F'F-F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 93 "de := diff(x(t),t$2)+10000*x(t)=800*cos(100* t);\nic := x(0)=0,D(x)(0)=0;\ndsolve(\{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\" \"\"*&\"&++\"F2F*F2F2,$*&\"$+)F2-%$cosG6#,$*&\"$+\"F2F-F2F2F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F) F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*(\"\"%\"\"\"F'F +-%$sinG6#,$*&\"$+\"F+F'F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "We could consider another similar initial value problem \+ where the frequency of the forcing function is even closer to the natu ral frequency. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^ 2*x/(d*t^2)+10000*x = 800*cos*99*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F& F)*$%\"tGF'F)!\"\"F)*&\"&++\"F)F(F)F)**\"$+)F)%$cosGF)\"#**F)F,F)" } {TEXT -1 4 ", " }{XPPEDIT 18 0 " x(0) = 0" "6#/-%\"xG6#\"\"!F'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0" "6#/-%%evalG6$ *&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The solution is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x(t)=-800/199" "6#/-%\"xG6#%\"tG,$*&\"$+)\"\"\"\"$*>!\" \"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*100*t +800/199" "6#,&*(%$cos G\"\"\"\"$+\"F&%\"tGF&F&*&\"$+)F&\"$*>!\"\"F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "cos*99*t)" "6#*(%$cosG\"\"\"\"#**F%%\"tGF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=1600/199" "6#/-%\"xG6#%\"tG*&\"%+;\"\"\"\"$ *>!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin( t/2)* sin(199/2* t)" "6# *&-%$sinG6#*&%\"tG\"\"\"\"\"#!\"\"F)-F%6#*(\"$*>F)F*F+F(F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "de := diff(x(t),t$2)+10000*x(t)=800*cos(99*t);\nic := x(0)=0,D(x)(0)=0;\ndsolve(\{de,ic\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*& \"&++\"F2F*F2F2,$*&\"$+)F2-%$cosG6#,$*&\"#**F2F-F2F2F2F2" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&#\"$+)\"$*>\"\"\"-%$ cosG6#,$*&\"$+\"F-F'F-F-F-!\"\"*&#F+F,F--F/6#,$*&\"#**F-F'F-F-F-F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "1600/199*sin(t/2)*sin(199/2*t);\ncombine(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&#\"%+;\"$*>\"\"\"*&-%$sinG6#,$*&\"\"#!\"\"%\"tGF( F(F(-F+6#,$*(F'F(F/F0F1F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*&#\"$+)\"$*>\"\"\"-%$cosG6#,$*&\"$+\"F(%\"tGF(F(F(!\"\"*&#F&F'F(-F*6 #,$*&\"#**F(F/F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "As before, the new solution " }{XPPEDIT 18 0 "k(t) = 1600/199;" "6#/-%\"kG6#%\"tG*&\"%+;\"\"\"\"$*>!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin( t/2)* sin(199/2* t)" "6#*&-%$sinG6#*&%\"tG\"\"\" \"\"#!\"\"F)-F%6#*(\"$*>F)F*F+F(F)F)" }{TEXT -1 56 " can be compared g raphically with the resonant solution " }{XPPEDIT 18 0 "g(t) = 4*t*sin *100*t;" "6#/-%\"gG6#%\"tG*,\"\"%\"\"\"F'F*%$sinGF*\"$+\"F*F'F*" } {TEXT -1 2 ". 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>!>#QH#F^z7$$\"3**H&e'\\fv.IF^z$!3YxpZ$Gq$)G$F^z7$Ff]bl$!3uY39YNYDUF^z 7$F[^bl$!3EgHf'['p)3&F^z7$F`^bl$!3'*[2*\\4bH'eF^z7$Fe^bl$!3'3Bno?\"oMl F^z7$Fj^bl$!3%)yH0Y*4@4(F^z7$Fd_bl$!3d^s#pr$=FyF^z7$F^`bl$!3KGerhi&o,) F^z7$Fh`bl$!3)=![YI=?)p(F^z7$Fbabl$!36\"*)*HQ]0EpF^z7$Fgabl$!3mNM8zO8$ Q'F^z7$F\\bbl$!3'4'=#ewJeu&F^z7$Fabbl$!3]^:r=?cB]F^z7$Ffbbl$!3sZ$Hfw$* pA%F^z7$$\"3[a)>`&GtEIF^z$!3#)Hu!3K&*yO$F^z7$F[cbl$!3%*y\\1-,'*eCF^z7$ $\"3+M,YSzO^\"F^z7$F`cbl$!35*)**[MAUeaF@7$Fecbl$\"3!** [E[Fy1I%F@7$Fjcbl$\"3O'fF^z7$Fhebl$\"3#[**G&G7#=l'F^z7$Fbfbl$\"3MgK12j^LwF^z7$F \\gbl$\"3jX.(oHOc-)F^z7$Ffgbl$\"3i1s(Qg2yz(F^z7$F`hbl$\"3[(z$*Q**\\v'p F^z7$Fehbl$\"3p'>c+hC[M'F^z7$Fjhbl$\"33D\"4[BH*)f&F^z7$F_ibl$\"3s&Q2) \\BLWZF^z7$Fdibl$\"3aL1MbIh(z$F^z7$$\"3idY,EE'*eIF^z$\"3G&y\\D^\"z()GF ^z7$Fiibl$\"3eK?'*fi2L>F^z7$$\"3!\\nYa2p91$F^z$\"37#*4'o)=#H[*F@7$F^jb l$!3xO6q$o/o7&F-7$$\"3=#py[_vR1$F^z$!3)yl*\\J:3]5F^z7$Fcjbl$!3+uYr!=OE .#F^z7$$\"3Y42Ju>[mIF^z$!35))QHv;n$)HF^z7$Fhjbl$!3_V_T(\\;%))QF^z7$F][ cl$!3^`LZ&Q\"ez=(F^z7$Fi^cl$!3UhF$ec>\"[mF^z 7$F^_cl$!3Yi?e[[\"4*fF^z7$Fc_cl$!3[!pH'*RSzA&F^z7$Fh_cl$!3U;VK0-msVF^z 7$$\"3W\"pqry'y*3$F^z$!3F>-#y=o,W$F^z7$F]`cl$!37f#>=!=#pW#F^z7$$\"3o#* 3/&=fC4$F^z$!3b>4T\\:X59F^z7$Fb`cl$!3[Qm$yS:0\\$F@7$$\"3!R46HeJ^4$F^z$ \"3M(*H!eOMa=(F@7$Fg`cl$\"3_Xv@R')[tk%F^z7$Ffacl$\"3 \"f[\"\\T`NPaF^z7$F[bcl$\"3xu[jc?&[9'F^z7$F`bcl$\"3-4hHeH*Hv'F^z7$Fjbc l$\"3([/tBdAOj(F^z7$Fdccl$\"3?A$G@c&[A!)F^z7$F^dcl$\"3I#*yANm]%*yF^z7$ Fhdcl$\"3y*zwS$4\"zD(F^z7$F]ecl$\"3m9w`4:ZgnF^z7$Fbecl$\"3!Gh&**f`p`hF ^z7$Fgecl$\"3=;%Qe$QRZaF^z7$F\\fcl$\"3)[mm-5))Hl%F^z7$$\"3I]xF_r(47$F^ z$\"3.q3\\(p!y:PF^z7$Fafcl$\"3%o)**[`$z\"4FF^z7$$\"3s$Qn`)esBJF^z$\"3% Q!Q_8#z>l\"F^z7$Fffcl$\"3tBH%QT0#RcF@7$$\"38 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 289 104 "An under-damped system for which the fr equency of the forcing term coincides with the natural frequency " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 7 "Example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 41 "Consider the two initial value problems : " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) +9*x = 5*cos*3*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\" F)*&\"\"*F)F(F)F)**\"\"&F)%$cosGF)\"\"$F)F,F)" }{TEXT -1 15 ", x(0) \+ = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG \"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d ^2*x/(d*t^2)+1/60 " "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\" F(*&F(F(\"#gF,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+9*x = 5*cos*3* t;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&\"\"*F'%\"xGF'F'**\"\"&F'%$cosGF' \"\"$F'%\"tGF'" }{TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx /dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The first initial value problem was considered previously and has the solution " }{XPPEDIT 18 0 "x(t) = 5/6;" "6#/-%\"xG6#%\"tG *&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t*sin*3*t" "6 #**%\"tG\"\"\"%$sinGF%\"\"$F%F$F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 211 "This solution exhibits the phomena of resonance in a sit uation where the frequency of the forcing term coincides with the natu ral frequency associated with the corresponding homogeneous simple har monic equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^ 2*x/(d*t^2)+9*x = 0" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\" \"F)*&\"\"*F)F(F)F)\"\"!" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 122 "which is the natural frequency of a possible underlying physic al oscillatory system modelled by the differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The \"extra\" t erm " }{XPPEDIT 18 0 "1/60" "6#*&\"\"\"F$\"#g!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 151 ", whi ch is introduced to form the second differential equation, could corre spond to the presence of a small amount of damping in an underlying sy stem. " }}{PARA 0 "" 0 "" {TEXT -1 89 "To solve the second initial val ue problem consider the homogeneous differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+1/60 " "6#,&* (%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(\"#gF,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+9*x = 0;" "6#/,&*&%#dxG\"\"\"%#dtG!\"\" F'*&\"\"*F'%\"xGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+m/60+9=0" "6#/,(*$%\"mG\"\"#\"\"\"*&F&F(\"#g!\"\"F (\"\"*F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Complet ing the square gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+m/60+1/14400=-9+1/14400" "6#/,(*$%\"mG\"\"#\"\"\"*&F&F(\"#g! \"\"F(*&F(F(\"&+W\"F+F(,&\"\"*F+*&F(F(F-F+F(" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+1/120)^2=-129599/14400" "6#/*$,&%\"mG\"\"\"*&F' F'\"$?\"!\"\"F'\"\"#,$*&\"'*fH\"F'\"&+W\"F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "m+1/120=``" "6#/,&%\"mG\"\"\"*&F&F&\"$?\"!\"\"F&%!G " }{TEXT 294 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(129599)/120;" "6#*&-%%sqrtG6#\"'*fH\"\"\"\"\"$?\"!\"\"" }{TEXT -1 1 " " }{TEXT 301 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m=-1/120" "6#/%\"mG,$ *&\"\"\"F'\"$?\"!\"\"F)" }{TEXT -1 1 " " }{TEXT 295 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "19*sqrt(359)/120;" "6#*(\"#>\"\"\"-%%sqrtG6#\"$f$F %\"$?\"!\"\"" }{TEXT -1 1 " " }{TEXT 302 1 "i" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The compl ementary solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c(t)=C[1]*exp(-t/120)*cos(19*sqrt(359)/120*t) + C[2]*exp(-t/120) *sin(19*sqrt(359)/120*t)" "6#/-%\"cG6#%\"tG,&*(&%\"CG6#\"\"\"F--%$expG 6#,$*&F'F-\"$?\"!\"\"F4F--%$cosG6#**\"#>F--%%sqrtG6#\"$f$F-F3F4F'F-F-F -*(&F+6#\"\"#F--F/6#,$*&F'F-F3F4F4F--%$sinG6#**F9F--F;6#F=F-F3F4F'F-F- F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "The appropriate form for a trial particular solution is : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/% \"xG-%\"pG6#%\"tG" }{XPPEDIT 18 0 "`` = A*cos*3*t+B*sin*3*t;" "6#/%!G, &**%\"AG\"\"\"%$cosGF(\"\"$F(%\"tGF(F(**%\"BGF(%$sinGF(F*F(F+F(F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=``" "6#/*&%#dxG\"\"\"%#dtG!\" \"%!G" }{XPPEDIT 18 0 "`p '`(t) = -3*A*sin*3*t+3*B*cos*3*t;" "6#/-%$p~ 'G6#%\"tG,&*,\"\"$\"\"\"%\"AGF+%$sinGF+F*F+F'F+!\"\"*,F*F+%\"BGF+%$cos GF+F*F+F'F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)=``" "6# /*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"%!G" }{XPPEDIT 18 0 "`p \+ ''`(t) = -9*A*cos*3*t-9*B*sin*3*t;" "6#/-%%p~''G6#%\"tG,&*,\"\"*\"\"\" %\"AGF+%$cosGF+\"\"$F+F'F+!\"\"*,F*F+%\"BGF+%$sinGF+F.F+F'F+F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " } {XPPEDIT 18 0 "d^2*x/(d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF %F'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dt G!\"\"" }{TEXT -1 5 " and " }{TEXT 293 1 "x" }{TEXT -1 44 " in the sec ond differential equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-9*A*cos*3*t-9*B*sin*3*t +1/60" "6#,(*,\"\"*\"\"\"%\"A GF&%$cosGF&\"\"$F&%\"tGF&!\"\"*,F%F&%\"BGF&%$sinGF&F)F&F*F&F+*&F&F&\"# gF+F&" }{XPPEDIT 18 0 " ``(-3*A*sin*3*t+3*B*cos*3*t) +9*(A*cos*3*t+B*s in*3*t)=5*cos*3*t" "6#/,&-%!G6#,&*,\"\"$\"\"\"%\"AGF+%$sinGF+F*F+%\"tG F+!\"\"*,F*F+%\"BGF+%$cosGF+F*F+F.F+F+F+*&\"\"*F+,&**F,F+F2F+F*F+F.F+F +**F1F+F-F+F*F+F.F+F+F+F+**\"\"&F+F2F+F*F+F.F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "By compar ing coefficients of the cosine and sine terms we obtain the equations: " }{XPPEDIT 18 0 "-A/20 = 0;" "6#/,$*&%\"AG\"\"\"\"#?!\"\"F)\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B/20 = 5;" "6#/*&%\"BG\"\"\"\"#?!\" \"\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } {XPPEDIT 18 0 "A = 0;" "6#/%\"AG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B = 100;" "6#/%\"BG\"$+\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular solution " }{XPPEDIT 18 0 " p(t) = 100*sin*3*t;" "6#/-%\"pG6#%\"tG**\"$+\"\"\"\"%$sinGF*\"\"$F*F'F *" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "general solution" }{TEXT -1 47 " for the second differential equation given by " }{XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6 #F'\"\"\"-%\"pG6#F'F," }{TEXT -1 14 " is therefore " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = C[1]*exp(-t/120)*cos(19*sqrt (359)/120*t)+C[2]*exp(-t/120)*sin(19*sqrt(359)/120*t)+100*sin*3*t" "6# /-%\"xG6#%\"tG,(*(&%\"CG6#\"\"\"F--%$expG6#,$*&F'F-\"$?\"!\"\"F4F--%$c osG6#**\"#>F--%%sqrtG6#\"$f$F-F3F4F'F-F-F-*(&F+6#\"\"#F--F/6#,$*&F'F-F 3F4F4F--%$sinG6#**F9F--F;6#F=F-F3F4F'F-F-F-**\"$+\"F-FGF-\"\"$F-F'F-F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "x(0)=0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 18 ", it follows that " } {XPPEDIT 18 0 "C[1]=0" "6#/&%\"CG6#\"\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x = C[2]*exp(-t/120)*sin(19*sqrt(359)/120*t)+100*sin *3*t" "6#/%\"xG,&*(&%\"CG6#\"\"#\"\"\"-%$expG6#,$*&%\"tGF+\"$?\"!\"\"F 3F+-%$sinG6#**\"#>F+-%%sqrtG6#\"$f$F+F2F3F1F+F+F+**\"$+\"F+F5F+\"\"$F+ F1F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "we have" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt =-C[2]/120" "6# /*&%#dxG\"\"\"%#dtG!\"\",$*&&%\"CG6#\"\"#F&\"$?\"F(F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-t/120)*sin(19*sqrt(359)/120*t)+19*sqrt(359)/120 *C[2]" "6#,&*&-%$expG6#,$*&%\"tG\"\"\"\"$?\"!\"\"F-F+-%$sinG6#**\"#>F+ -%%sqrtG6#\"$f$F+F,F-F*F+F+F+**F2F+-F46#F6F+F,F-&%\"CG6#\"\"#F+F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/120)*cos(19*sqrt(359)/120*t) +30 0*cos*3*t" "6#,&*&-%$expG6#,$*&%\"tG\"\"\"\"$?\"!\"\"F-F+-%$cosG6#**\" #>F+-%%sqrtG6#\"$f$F+F,F-F*F+F+F+**\"$+$F+F/F+\"\"$F+F*F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Eval(dx/dt,t=0)=0" "6#/-%%EvalG6$*&%#dxG\" \"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 17 ", it follws that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "19*sqrt(359)/120*C[2]+300=0 " "6#/,&**\"#>\"\"\"-%%sqrtG6#\"$f$F'\"$?\"!\"\"&%\"CG6#\"\"#F'F'\"$+$ F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C[2]=-36000/(19*sqrt( 359)" "6#/&%\"CG6#\"\"#,$*&\"&+g$\"\"\"*&\"#>F+-%%sqrtG6#\"$f$F+!\"\"F 2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "Hence the solution \+ to the second initial value problem is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 100*sin*3*t-36000/(19*sqrt(359));" "6# /-%\"xG6#%\"tG,&**\"$+\"\"\"\"%$sinGF+\"\"$F+F'F+F+*&\"&+g$F+*&\"#>F+- %%sqrtG6#\"$f$F+!\"\"F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t/120)*s in(19*sqrt(359)/120*t);" "6#*&-%$expG6#,$*&%\"tG\"\"\"\"$?\"!\"\"F,F*- %$sinG6#**\"#>F*-%%sqrtG6#\"$f$F*F+F,F)F*F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 296 31 "____________________________ ___" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 14 "transient part" }{TEXT -1 21 " of the solution is " }{XPPEDIT 18 0 "q(t) = -36000/(19*sqrt(359));" "6#/-%\"qG6#%\"tG,$*&\"&+g$\"\"\"*& \"#>F+-%%sqrtG6#\"$f$F+!\"\"F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-t /120)*sin(19*sqrt(359)/120*t);" "6#*&-%$expG6#,$*&%\"tG\"\"\"\"$?\"!\" \"F,F*-%$sinG6#**\"#>F*-%%sqrtG6#\"$f$F*F+F,F)F*F*" }{TEXT -1 13 ", w hile the " }{TEXT 259 12 "steady-state" }{TEXT -1 10 " part is " } {XPPEDIT 18 0 "p(t)=100*sin*3*t" "6#/-%\"pG6#%\"tG**\"$+\"\"\"\"%$sinG F*\"\"$F*F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Solution using " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "de := diff(x(t),t$2)+1/60*d iff(x(t),t)+9*x(t)=5*cos(3*t);\nic := x(0)=0,D(x)(0)=0;\ndesolve(\{de, ic\},x(t),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%d iffG6$-%\"xG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&#F2\"#gF2-F(6$F*F-F2F2*&\"\" *F2F*F2F2,$*&\"\"&F2-%$cosG6#,$*&\"\"$F2F-F2F2F2F2" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG6$/-%\"xG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*&\"#g\"\"\")%\"mG \"\"#F(F(F*F(\"$S&F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G,&#\"\"\"\"$?\"!\"\"*&^## \"#>F'F&\"$f$#F&\"\"#F&,&#F&F'F(*&^##!#>F'F&F-F.F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F)F/F0\"$f$#F)\"\"#F1F)F)F)F)*(&F'6#F:F)F*F)-%$cosGF 4F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%9particular~solution~.~.~G,&*(-%$expG6#,$*&\"$?\"!\"\" %\"tG\"\"\"F,F.-%$sinG6#,$**\"#>F.F+F,\"$f$#F.\"\"#F-F.F.F.-%$IntG6$,& *&#\"$+$\"%@oF.*(F5F6-F'6#,$*&F+F,F-F.F.F.-%$cosG6#,&*&\"\"$F.F-F.F,*& #F4F+F.*&F5F6F-F.F.F.F.F.F.*&F=F.*(F5F6FAF.-FF6#,&*&FJF.F-F.F.*&FLF.FM F.F.F.F.F.F-F.F.*(F&F.-FFF1F.-F96$,&*&#F>F?F.*(F5F6FAF.-F0FQF.F.F,*&#F >F?F.*(F5F6FAF.-F0FGF.F.F,F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%) ~~~~~~~=G,$*&\"$+\"\"\"\"-%$sinG6#,$*&\"\"$F'%\"tGF'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6gen eral~solution~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG, (*&\"$+\"\"\"\"-%$sinG6#,$*&\"\"$F+F'F+F+F+F+*(&%\"CG6#F+F+-%$expG6#,$ *&\"$?\"!\"\"F'F+FF+F;F<\"$f$#F+\"\"#F'F+F+F+F+*(&F46# FDF+F6F+-%$cosGF>F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afrom~the~initial~conditions~.~.~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!&%\"CG6#\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"\"!,(\"$+$\"\"\"*&#\"#>\"$?\"F'*&&%\"CG6#F'F'\"$f$ #F'\"\"#F'F'*&#F'F+F'&F.6#F2F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.~.~G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\",$*(\"&+g$F'\"%@o!\"\"\"$f$#F'\" \"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" xG6#%\"tG,&*&\"$+\"\"\"\"-%$sinG6#,$*&\"\"$F+F'F+F+F+F+*&#\"&+g$\"%@oF +*(-%$expG6#,$*&\"$?\"!\"\"F'F+F=F+-F-6#,$**\"#>F+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 73 "Note that the angular frequency of the transient p art of the solution is " }{XPPEDIT 18 0 "19*sqrt(359)/120" "6#*(\"#>\" \"\"-%%sqrtG6#\"$f$F%\"$?\"!\"\"" }{TEXT -1 1 " " }{TEXT 297 1 "~" } {TEXT -1 98 " 2.999988426, which is close to the angular frequency 3 \+ of the steady-state part of the solution." }}{PARA 0 "" 0 "" {TEXT -1 10 "Also when " }{XPPEDIT 18 0 "t =0" "6#/%\"tG\"\"!" }{TEXT -1 57 ", \+ the amplitude of the transient part of the solution is " }{XPPEDIT 18 0 "36000/(19*sqrt(359))" "6#*&\"&+g$\"\"\"*&\"#>F%-%%sqrtG6#\"$f$F%!\" \"" }{TEXT -1 1 " " }{TEXT 298 1 "~" }{TEXT -1 31 " 100.0003858. Since the factor " }{XPPEDIT 18 0 "exp(-t/120)" "6#-%$expG6#,$*&%\"tG\"\"\" \"$?\"!\"\"F+" }{TEXT -1 33 " only decreases slowly from 1 as " } {TEXT 299 1 "t" }{TEXT -1 153 " increases from 0, the amplitude of the transient part of the solution initially remains close to amplitude 1 00 of the steady-state part of the solution." }}{PARA 0 "" 0 "" {TEXT -1 80 "Initially the transient and steady state terms almost cancel ou t, and only when " }{TEXT 300 1 "t" }{TEXT -1 76 " reaches quite large values does the transient part effectively decay away. 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" }}{PARA 0 "" 0 "" {TEXT -1 165 "Initially there is l ittle difference between the two solutions as can be seen by the follo wing picture in which the two solutions are plotted over the interval \+ from " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "t=20" "6#/%\"tG\"#?" }{TEXT -1 2 ". 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'R7Qd1.$F`_l7$F[gw$!3yI!z#4h0NFF`_l7$F`gw$!3[Ab:>\\%>S#F`_l7$Fegw$!3/j #HC\\;e.#F`_l7$Fjgw$!31!GhQL<;k\"F`_l7$F_hw$!3Fgn#>%4oC7F`_l7$Fdhw$!3R t(3([kb1zFB7$Fihw$!3yw8K(y!GaMFB7$F^iw$\"33/yHflc\\5FB7$Fciw$\"3_L<*fv qPa&FB7$Fhiw$\"3;wk$4>&=n**FB7$F]jw$\"3\"3/hbMcfU\"F`_l7$Fbjw$\"3m%H=J *zBO=F`_l7$Fgjw$\"3+&=Llf:w]#F`_l7$F\\[x$\"3CCV=W@aqIF`_l7$Fa[x$\"3IIk .NR>+NF`_l7$Ff[x$\"3ui>[%)>`xPF`_l7$F[\\x$\"3-t^'3\")G\\&QF`_l7$F`\\x$ \"3A7P6ob9!*QF`_l7$Fe\\x$\"3y]F)pa&y\"*QF`_l7$Fj\\x$\"3I,+S4xv#)QF`_l7 $F_]x$\"3))R(4q$>3jQF`_l7$Fd]x$\"3@*zmLJ2G$QF`_l7$Fi]x$\"3@%>@O-:xg$F` _l7$F^^x$\"3Q,vj\"34WA$F`_l7$Fc^x$\"3F0oo[EMUHF`_l7$Fh^x$\"3X,En()=S=E F`_l7$F]_x$\"3%=xr,0ArD#F`_l7$Fb_x$\"3m3HSH3dj=F`_l7$Fg_x$\"3#*GvNvsFV 9F`_l7$F\\`x$\"3Zl-&\\6a@+\"F`_l7$Fa`x$\"3?W@k&oXTY&FB7$Ff`x$\"3ePp8IL [[#)F-7$F[ax$!3`Wy!HMl3$QFB7$F`ax$!3cWlU\"e5sV)FB7$Feax$!3E&*)z-I.HH\" F`_l7$Fjax$!3bFt`AIFC " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "General case" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 44 "Consider the genera l initial value problem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*& F%F(*$%\"tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = F[0]*cos*omega*t;" "6#/,&*&%#dxG\"\"\"%#dtG!\" \"F'*&%&omegaG\"\"#%\"xGF'F'**&%\"FG6#\"\"!F'%$cosGF'F+F'%\"tGF'" } {TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "lambda" "6#%'lamb daG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 18 " are positive and " }{XPPEDIT 18 0 "lambda < omega;" "6#2%'lamb daG%&omegaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "The corre sponding homogeneous differential equation: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^ 2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F& F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = 0; " "6#/,&*&%#dxG\"\"\"%#dtG!\"\"F'*&%&omegaG\"\"#%\"xGF'F'\"\"!" } {TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "has the auxiliary equa tion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+2*lambda *m+omega^2=0" "6#/,(*$%\"mG\"\"#\"\"\"*(F'F(%'lambdaGF(F&F(F(*$%&omega GF'F(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+lambda)^2=lambda^2 -omega^2" "6#/*$,&%\"mG\"\"\"%'lambdaGF'\"\"#,&*$F(F)F'*$%&omegaGF)!\" \"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "giving " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m=-lambda" "6#/%\"mG,$%'lam bdaG!\"\"" }{TEXT -1 1 " " }{TEXT 290 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(omega^2-lambda^2)*i;" "6#*&-%%sqrtG6#,&*$%&omegaG\"\"#\"\" \"*$%'lambdaGF*!\"\"F+%\"iGF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The complementary solutio n of the differential equation above is " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "c(t) = C[1]*exp(-lambda*t)*cos(sqrt(omega^2 -lambda^2)*t)+C[2]*exp(-lambda*t)*sin(sqrt(omega^2-lambda^2)*t);" "6#/ -%\"cG6#%\"tG,&*(&%\"CG6#\"\"\"F--%$expG6#,$*&%'lambdaGF-F'F-!\"\"F--% $cosG6#*&-%%sqrtG6#,&*$%&omegaG\"\"#F-*$F3F?F4F-F'F-F-F-*(&F+6#F?F--F/ 6#,$*&F3F-F'F-F4F--%$sinG6#*&-F:6#,&*$F>F?F-*$F3F?F4F-F'F-F-F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The appropriate trial particular solution for the non-hom egeneous differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*& F%F(*$%\"tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = F[0]*cos*omega*t;" "6#/,&*&%#dxG\"\"\"%#dtG!\" \"F'*&%&omegaG\"\"#%\"xGF'F'**&%\"FG6#\"\"!F'%$cosGF'F+F'%\"tGF'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "is: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x = p(t)" "6#/%\"xG-%\"pG6#%\"tG" } {XPPEDIT 18 0 "`` = A*cos*omega*t+B*sin*omega*t;" "6#/%!G,&**%\"AG\"\" \"%$cosGF(%&omegaGF(%\"tGF(F(**%\"BGF(%$sinGF(F*F(F+F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = `p '`(t);" "6#/*&%#dxG\"\"\"%#dtG!\" \"-%$p~'G6#%\"tG" }{XPPEDIT 18 0 "`` = -A*omega*sin*omega*t+B*omega*co s*omega*t;" "6#/%!G,&*,%\"AG\"\"\"%&omegaGF(%$sinGF(F)F(%\"tGF(!\"\"*, %\"BGF(F)F(%$cosGF(F)F(F+F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d ^2*x/(d*t^2) = `p ''`(t);" "6#/*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F (!\"\"-%%p~''G6#F+" }{XPPEDIT 18 0 "`` = -A*omega^2*cos*omega*t-B*omeg a^2*sin*omega*t;" "6#/%!G,&*,%\"AG\"\"\"*$%&omegaG\"\"#F(%$cosGF(F*F(% \"tGF(!\"\"*,%\"BGF(*$F*F+F(%$sinGF(F*F(F-F(F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 17 "Substituting for " }{XPPEDIT 18 0 "d^2*x/ (d*t^2)" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dx/dt" "6#*&%#dxG\"\"\"%#dtG!\"\"" }{TEXT -1 5 " and " }{TEXT 291 1 "x" }{TEXT -1 37 " in the differential equation g ives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-A*omega^2*cos*omega*t-B*omega^2*sin*omega*t +2* lambda*(-A*omega*sin*omega*t+B*omega*cos*omega*t)+omega^2*(A*cos*omega *t+B*sin*omega*t)=F[0]*cos(omega*t)" "6#/,**,%\"AG\"\"\"*$%&omegaG\"\" #F'%$cosGF'F)F'%\"tGF'!\"\"*,%\"BGF'*$F)F*F'%$sinGF'F)F'F,F'F-*(F*F'%' lambdaGF',&*,F&F'F)F'F1F'F)F'F,F'F-*,F/F'F)F'F+F'F)F'F,F'F'F'F'*&F)F*, &**F&F'F+F'F)F'F,F'F'**F/F'F1F'F)F'F,F'F'F'F'*&&%\"FG6#\"\"!F'-F+6#*&F )F'F,F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "By comparin g coefficients of the cosine and sine terms we obtain the equations: \+ " }{XPPEDIT 18 0 "-2*A*lambda = 0;" "6#/,$*(\"\"#\"\"\"%\"AGF'%'lambda GF'!\"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2*B*lambda*omega = \+ F[0];" "6#/**\"\"#\"\"\"%\"BGF&%'lambdaGF&%&omegaGF&&%\"FG6#\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 " A = 0;" "6#/%\"AG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B = F[0]/( 2*lambda*omega);" "6#/%\"BG*&&%\"FG6#\"\"!\"\"\"*(\"\"#F*%'lambdaGF*%& omegaGF*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "Thus we obtain the particular solution " }{XPPEDIT 18 0 "p(t) = F[0]*sin*omeg a*t/(2*lambda*omega);" "6#/-%\"pG6#%\"tG*,&%\"FG6#\"\"!\"\"\"%$sinGF-% &omegaGF-F'F-*(\"\"#F-%'lambdaGF-F/F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "general solution" }{TEXT -1 47 " for the second differential equation given by " }{XPPEDIT 18 0 "x(t)=c(t)+p(t)" "6#/-%\"xG6#%\"tG,&-%\"cG6#F'\"\"\"-%\"pG6#F'F," } {TEXT -1 14 " is therefore " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = C[1]*exp(-lambda*t)* cos(sqrt(omega^2-lambda^2)*t)+C[2]*exp(-lambda*t)*sin(sqrt(omega^2-lam bda^2)*t)+F[0]*sin*omega*t/(2*lambda*omega);" "6#/-%\"xG6#%\"tG,(*(&% \"CG6#\"\"\"F--%$expG6#,$*&%'lambdaGF-F'F-!\"\"F--%$cosG6#*&-%%sqrtG6# ,&*$%&omegaG\"\"#F-*$F3F?F4F-F'F-F-F-*(&F+6#F?F--F/6#,$*&F3F-F'F-F4F-- %$sinG6#*&-F:6#,&*$F>F?F-*$F3F?F4F-F'F-F-F-*,&%\"FG6#\"\"!F-FIF-F>F-F' F-*(F?F-F3F-F>F-F4F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 " Since " }{XPPEDIT 18 0 "x(0)=0" "6#/-%\"xG6#\"\"!F'" }{TEXT -1 18 ", i t follows that " }{XPPEDIT 18 0 "C[1]=0" "6#/&%\"CG6#\"\"\"\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = C[2]*exp(-lambda*t)*sin(sqrt(o mega^2-lambda^2)*t)+F[0]*sin*omega*t/(2*lambda*omega);" "6#/%\"xG,&*(& %\"CG6#\"\"#\"\"\"-%$expG6#,$*&%'lambdaGF+%\"tGF+!\"\"F+-%$sinG6#*&-%% sqrtG6#,&*$%&omegaGF*F+*$F1F*F3F+F2F+F+F+*,&%\"FG6#\"\"!F+F5F+F=F+F2F+ *(F*F+F1F+F=F+F3F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "we \+ have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt = -C[2 ]*lambda*exp(-lambda*t)*sin(sqrt(omega^2-lambda^2)*t)+C[2]*sqrt(omega^ 2-lambda^2)*exp(-lambda*t)*cos(sqrt(omega^2-lambda^2)*t)+F[0]*cos*omeg a*t/(2*lambda);" "6#/*&%#dxG\"\"\"%#dtG!\"\",(**&%\"CG6#\"\"#F&%'lambd aGF&-%$expG6#,$*&F/F&%\"tGF&F(F&-%$sinG6#*&-%%sqrtG6#,&*$%&omegaGF.F&* $F/F.F(F&F5F&F&F(**&F,6#F.F&-F;6#,&*$F?F.F&*$F/F.F(F&-F16#,$*&F/F&F5F& F(F&-%$cosG6#*&-F;6#,&*$F?F.F&*$F/F.F(F&F5F&F&F&*,&%\"FG6#\"\"!F&FNF&F ?F&F5F&*&F.F&F/F&F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Eval(dx/dt,t=0 )=0" "6#/-%%EvalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 17 ", it follws that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C[2]*sqrt(omega^2-lambda^2)+F[0]/(2*lambda) = 0;" "6#/,&*&&%\"CG6#\" \"#\"\"\"-%%sqrtG6#,&*$%&omegaGF)F**$%'lambdaGF)!\"\"F*F**&&%\"FG6#\" \"!F**&F)F*F2F*F3F*F8" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C[2] = -F [0]/(2*lambda*sqrt(omega^2-lambda^2));" "6#/&%\"CG6#\"\"#,$*&&%\"FG6# \"\"!\"\"\"*(F'F.%'lambdaGF.-%%sqrtG6#,&*$%&omegaGF'F.*$F0F'!\"\"F.F8F 8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The solution to the initial value problem is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x(t) = F[0]*sin*omega*t/(2*lambda*omega)-F[0]/(2*lambda *sqrt(omega^2-lambda^2));" "6#/-%\"xG6#%\"tG,&*,&%\"FG6#\"\"!\"\"\"%$s inGF.%&omegaGF.F'F.*(\"\"#F.%'lambdaGF.F0F.!\"\"F.*&&F+6#F-F.*(F2F.F3F .-%%sqrtG6#,&*$F0F2F.*$F3F2F4F.F4F4" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e xp(-lambda*t)*sin(sqrt(omega^2-lambda^2)*t)" "6#*&-%$expG6#,$*&%'lambd aG\"\"\"%\"tGF*!\"\"F*-%$sinG6#*&-%%sqrtG6#,&*$%&omegaG\"\"#F**$F)F7F, F*F+F*F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 292 33 "_________________________________" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 14 "transient part" }{TEXT -1 20 " of the solution is " }{XPPEDIT 18 0 "q(t) = -F[0]/(2*lambda*sqrt(omega^2-lambda^2));" "6#/-%\"qG6#%\" tG,$*&&%\"FG6#\"\"!\"\"\"*(\"\"#F.%'lambdaGF.-%%sqrtG6#,&*$%&omegaGF0F .*$F1F0!\"\"F.F9F9" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-lambda*t)*sin (sqrt(omega^2-lambda^2)*t)" "6#*&-%$expG6#,$*&%'lambdaG\"\"\"%\"tGF*! \"\"F*-%$sinG6#*&-%%sqrtG6#,&*$%&omegaG\"\"#F**$F)F7F,F*F+F*F*" } {TEXT -1 13 ", while the " }{TEXT 259 12 "steady-state" }{TEXT -1 10 " part is " }{XPPEDIT 18 0 "p(t) = F[0]*sin*omega*t/(2*lambda*omega); " "6#/-%\"pG6#%\"tG*,&%\"FG6#\"\"!\"\"\"%$sinGF-%&omegaGF-F'F-*(\"\"#F -%'lambdaGF-F/F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Note that the amplitude " }{XPPEDIT 18 0 "F[0]/(2*lambda*omega)" "6#* &&%\"FG6#\"\"!\"\"\"*(\"\"#F(%'lambdaGF(%&omegaGF(!\"\"" }{TEXT -1 84 " of the steady-state part of the solution is inversely proportional t o the constant " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 34 " \+ associated with the damping term." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x(t) = F[0]/2;" "6#/-%\"xG6#%\"tG*&&%\"FG6#\"\"!\"\"\" \"\"#!\"\"" }{XPPEDIT 18 0 "``(sin*omega*t/(lambda*omega)-exp(-lambda* t)*sin(sqrt(omega^2-lambda^2)*t)/(lambda*sqrt(omega^2-lambda^2)));" "6 #-%!G6#,&**%$sinG\"\"\"%&omegaGF)%\"tGF)*&%'lambdaGF)F*F)!\"\"F)*(-%$e xpG6#,$*&F-F)F+F)F.F)-F(6#*&-%%sqrtG6#,&*$F*\"\"#F)*$F-F=F.F)F+F)F)*&F -F)-F96#,&*$F*F=F)*$F-F=F.F)F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "It is to be expected that this solution tends to the solution " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = F[0]*t*sin*omega*t/(2*omega);" "6#/-%\"xG6# %\"tG*.&%\"FG6#\"\"!\"\"\"F'F-%$sinGF-%&omegaGF-F'F-*&\"\"#F-F/F-!\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "of the initial probl em: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2) +omega^2*x = F[0]*cos*omega*t;" "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$% \"tGF'F)!\"\"F)*&%&omegaGF'F(F)F)**&%\"FG6#\"\"!F)%$cosGF)F/F)F,F)" } {TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 3 "as " }{XPPEDIT 18 0 "lambda->0" "6#f*6#%' lambdaG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 42 "To demonstrate this we need to check that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(sin*ome ga*t/(lambda*omega)-exp(-lambda*t)*sin(sqrt(omega^2-lambda^2)*t)/(lamb da*sqrt(omega^2-lambda^2))),lambda = 0) = t*sin*omega*t/omega;" "6#/-% &LimitG6$-%!G6#,&**%$sinG\"\"\"%&omegaGF-%\"tGF-*&%'lambdaGF-F.F-!\"\" F-*(-%$expG6#,$*&F1F-F/F-F2F--F,6#*&-%%sqrtG6#,&*$F.\"\"#F-*$F1FAF2F-F /F-F-*&F1F--F=6#,&*$F.FAF-*$F1FAF2F-F2F2/F1\"\"!*,F/F-F,F-F.F-F/F-F.F2 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "This follows from the series expansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*omega*t/(lambda*omega)-exp(-la mbda*t)*sin(sqrt(omega^2-lambda^2)*t)/(lambda*sqrt(omega^2-lambda^2)) \+ = ``(t*sin*omega*t/omega)-``((sin*omega*t-omega*t*cos*omega*t+omega^2* t^2*sin*omega*t)/(2*omega^3))*lambda+O(lambda^2);" "6#/,&**%$sinG\"\" \"%&omegaGF'%\"tGF'*&%'lambdaGF'F(F'!\"\"F'*(-%$expG6#,$*&F+F'F)F'F,F' -F&6#*&-%%sqrtG6#,&*$F(\"\"#F'*$F+F;F,F'F)F'F'*&F+F'-F76#,&*$F(F;F'*$F +F;F,F'F,F,,(-%!G6#*,F)F'F&F'F(F'F)F'F(F,F'*&-FE6#*&,(*(F&F'F(F'F)F'F' *,F(F'F)F'%$cosGF'F(F'F)F'F,*,F(F;F)F;F&F'F(F'F)F'F'F'*&F;F'*$F(\"\"$F 'F,F'F+F'F,-%\"OG6#*$F+F;F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "O(lambda^2)" "6#-%\"OG6#*$%'lambdaG\"\"# " }{TEXT -1 33 " is the \"tail\" of the series and " }{XPPEDIT 18 0 "L imit(O(lambda^2)/lambda,lambda = 0) = 0;" "6#/-%&LimitG6$*&-%\"OG6#*$% 'lambdaG\"\"#\"\"\"F,!\"\"/F,\"\"!F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 259 "" 0 "" {TEXT -1 44 "Details concerning the series e xpansion of " }{XPPEDIT 18 0 "sin*omega*t/(lambda*omega)-exp(-lambda* t)*sin(sqrt(omega^2-lambda^2)*t)/(lambda*sqrt(omega^2-lambda^2));" "6# ,&**%$sinG\"\"\"%&omegaGF&%\"tGF&*&%'lambdaGF&F'F&!\"\"F&*(-%$expG6#,$ *&F*F&F(F&F+F&-F%6#*&-%%sqrtG6#,&*$F'\"\"#F&*$F*F:F+F&F(F&F&*&F*F&-F66 #,&*$F'F:F&*$F*F:F+F&F+F+" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sqrt(omega^2-lambda^2) = omega*(1-lambda^2/omega^2)^(1/2)" "6#/- %%sqrtG6#,&*$%&omegaG\"\"#\"\"\"*$%'lambdaGF*!\"\"*&F)F+),&F+F+*&F-F** $F)F*F.F.*&F+F+F*F.F+" }{XPPEDIT 18 0 "``= omega*(1-lambda^2/(2*omega^ 2)) + O(lambda^4)" "6#/%!G,&*&%&omegaG\"\"\",&F(F(*&%'lambdaG\"\"#*&F, F(*$F'F,F(!\"\"F/F(F(-%\"OG6#*$F+\"\"%F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=omega+lambda^2/(2*omega) + O (lambda^4)" "6#/%!G,(%&omegaG\"\"\"*&%'lambdaG\"\"#*&F*F'F&F'!\"\"F'-% \"OG6#*$F)\"\"%F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sin(omega*t-lambda^2/(2*omega)*t+O(lambda^4)) = sin* omega*t*cos(lambda^2*t/(2*omega)+O(lambda^4))-cos*omega*t*sin(lambda^2 *t/(2*omega)+O(lambda^4));" "6#/-%$sinG6#,(*&%&omegaG\"\"\"%\"tGF*F**( %'lambdaG\"\"#*&F.F*F)F*!\"\"F+F*F0-%\"OG6#*$F-\"\"%F*,&**F%F*F)F*F+F* -%$cosG6#,&*(F-F.F+F**&F.F*F)F*F0F*-F26#*$F-F5F*F*F***F9F*F)F*F+F*-F%6 #,&*(F-F.F+F**&F.F*F)F*F0F*-F26#*$F-F5F*F*F0" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Since " } {XPPEDIT 18 0 "cos(lambda^2*t/(2*omega)+O(lambda^4)) = 1+O(lambda^4); " "6#/-%$cosG6#,&*(%'lambdaG\"\"#%\"tG\"\"\"*&F*F,%&omegaGF,!\"\"F,-% \"OG6#*$F)\"\"%F,,&F,F,-F16#*$F)F4F," }{TEXT -1 6 " and " }{XPPEDIT 18 0 "sin(lambda^2*t/(2*omega)+O(lambda^4)) = lambda^2*t/(2*omega)+O(l ambda^4);" "6#/-%$sinG6#,&*(%'lambdaG\"\"#%\"tG\"\"\"*&F*F,%&omegaGF,! \"\"F,-%\"OG6#*$F)\"\"%F,,&*(F)F*F+F,*&F*F,F.F,F/F,-F16#*$F)F4F," } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(sqrt(omega^2-lambda^2)*t) = sin*ome ga*t-``(t*cos*omega*t/(2*omega))*lambda^2+O(lambda^4);" "6#/-%$sinG6#* &-%%sqrtG6#,&*$%&omegaG\"\"#\"\"\"*$%'lambdaGF.!\"\"F/%\"tGF/,(*(F%F/F -F/F3F/F/*&-%!G6#*,F3F/%$cosGF/F-F/F3F/*&F.F/F-F/F2F/*$F1F.F/F2-%\"OG6 #*$F1\"\"%F/" }{TEXT -1 15 " ------- (i). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/sqrt(omega^2-lambda^2)=1/omega" "6#/* &\"\"\"F%-%%sqrtG6#,&*$%&omegaG\"\"#F%*$%'lambdaGF,!\"\"F/*&F%F%F+F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-lambda^2/(omega^2))^(-1/2)=1/omega " "6#/),&\"\"\"F&*&%'lambdaG\"\"#*$%&omegaGF)!\"\"F,,$*&F&F&F)F,F,*&F& F&F+F," }{XPPEDIT 18 0 " ``(1+lambda^2/(2*omega^2))+O(lambda^4)" "6#,& -%!G6#,&\"\"\"F(*&%'lambdaG\"\"#*&F+F(*$%&omegaGF+F(!\"\"F(F(-%\"OG6#* $F*\"\"%F(" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 19 "From (i) and (ii), " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sin(sqrt(omega^2-lambda^2)*t)/sqrt(omega^2-lambda^2)=si n*omega*t/omega+ ``((sin*omega*t-omega*t*cos*omega*t)/(2*omega^3) )*la mbda^2 +O(lambda^4)" "6#/*&-%$sinG6#*&-%%sqrtG6#,&*$%&omegaG\"\"#\"\" \"*$%'lambdaGF/!\"\"F0%\"tGF0F0-F*6#,&*$F.F/F0*$F2F/F3F3,(**F&F0F.F0F4 F0F.F3F0*&-%!G6#*&,&*(F&F0F.F0F4F0F0*,F.F0F4F0%$cosGF0F.F0F4F0F3F0*&F/ F0*$F.\"\"$F0F3F0*$F2F/F0F0-%\"OG6#*$F2\"\"%F0" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "exp(-lambda*t) = \+ 1-lambda*t+lambda^2*t^2/2+O(lambda^4);" "6#/-%$expG6#,$*&%'lambdaG\"\" \"%\"tGF*!\"\",*F*F**&F)F*F+F*F,*(F)\"\"#F+F0F0F,F*-%\"OG6#*$F)\"\"%F* " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-lambda*t)*sin(sqrt(omega^2-lambd a^2)*t)/sqrt(omega^2-lambda^2)=sin*omega*t/omega-``(t*sin*omega*t/omeg a)*lambda+``((sin*omega*t-omega*t*cos*omega*t)/(2*omega^3)+t^2*sin*ome ga*t/(2*omega))*lambda^2+O(lambda^3)" "6#/*(-%$expG6#,$*&%'lambdaG\"\" \"%\"tGF+!\"\"F+-%$sinG6#*&-%%sqrtG6#,&*$%&omegaG\"\"#F+*$F*F8F-F+F,F+ F+-F36#,&*$F7F8F+*$F*F8F-F-,***F/F+F7F+F,F+F7F-F+*&-%!G6#*,F,F+F/F+F7F +F,F+F7F-F+F*F+F-*&-FC6#,&*&,&*(F/F+F7F+F,F+F+*,F7F+F,F+%$cosGF+F7F+F, F+F-F+*&F8F+*$F7\"\"$F+F-F+*,F,F8F/F+F7F+F,F+*&F8F+F7F+F-F+F+*$F*F8F+F +-%\"OG6#*$F*FQF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin*omega*t/omeg a-``(t*sin*omega*t/omega)*lambda+``((sin*omega*t-omega*t*cos*omega*t+o mega^2*t^2*sin*omega*t)/(2*omega^3))*lambda^2+O(lambda^3);" "6#/%!G,** *%$sinG\"\"\"%&omegaGF(%\"tGF(F)!\"\"F(*&-F$6#*,F*F(F'F(F)F(F*F(F)F+F( %'lambdaGF(F+*&-F$6#*&,(*(F'F(F)F(F*F(F(*,F)F(F*F(%$cosGF(F)F(F*F(F+*, F)\"\"#F*F:F'F(F)F(F*F(F(F(*&F:F(*$F)\"\"$F(F+F(*$F0F:F(F(-%\"OG6#*$F0 F=F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*omega*t/(lambda*omega)- exp(-lambda*t)*sin(sqrt(omega^2-lambda^2)*t)/(lambda*sqrt(omega^2-lamb da^2)) = ``(t*sin*omega*t/omega)-``((sin*omega*t-omega*t*cos*omega*t+o mega^2*t^2*sin*omega*t)/(2*omega^3))*lambda+O(lambda^2);" "6#/,&**%$si nG\"\"\"%&omegaGF'%\"tGF'*&%'lambdaGF'F(F'!\"\"F'*(-%$expG6#,$*&F+F'F) F'F,F'-F&6#*&-%%sqrtG6#,&*$F(\"\"#F'*$F+F;F,F'F)F'F'*&F+F'-F76#,&*$F(F ;F'*$F+F;F,F'F,F,,(-%!G6#*,F)F'F&F'F(F'F)F'F(F,F'*&-FE6#*&,(*(F&F'F(F' F)F'F'*,F(F'F)F'%$cosGF'F(F'F)F'F,*,F(F;F)F;F&F'F(F'F)F'F'F'*&F;F'*$F( \"\"$F'F,F'F+F'F,-%\"OG6#*$F+F;F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(sin*omega*t/(lambda*omega)-exp(-lambda*t)*sin(sqrt(omega^2 -lambda^2)*t)/(lambda*sqrt(omega^2-lambda^2))),lambda = 0) = t*sin*ome ga*t/omega;" "6#/-%&LimitG6$-%!G6#,&**%$sinG\"\"\"%&omegaGF-%\"tGF-*&% 'lambdaGF-F.F-!\"\"F-*(-%$expG6#,$*&F1F-F/F-F2F--F,6#*&-%%sqrtG6#,&*$F .\"\"#F-*$F1FAF2F-F/F-F-*&F1F--F=6#,&*$F.FAF-*$F1FAF2F-F2F2/F1\"\"!*,F /F-F,F-F.F-F/F-F.F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "sin(omega*t)/(lambda*omega)-exp(-lambda*t)*sin(sqrt(omega^2-lam bda^2)*t)/(lambda*sqrt(omega^2-lambda^2));\n``=series(%,lambda,3):\n`` =simplify(rhs(%),symbolic);\nLimit(rhs(%),lambda=0);\n``=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%$sinG6#*&%&omegaG\"\"\"%\"tGF*F* %'lambdaG!\"\"F)F-F***-%$expG6#,$*&F,F*F+F*F-F*-F&6#*&,&*$)F)\"\"#F*F* *$)F,F:F*F-#F*F:F+F*F*F,F-F7#F-F:F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G+)%'lambdaG*(%\"tG\"\"\"-%$sinG6#*&%&omegaGF)F(F)F)F.!\"\"\"\"!,$ *&#F)\"\"#F)*&,(F*F)*(-%$cosGF,F)F(F)F.F)F/*()F(F4F)F*F))F.F4F)F)F)F.! \"$F)F/F)-%\"OG6#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$+) %'lambdaG*(%\"tG\"\"\"-%$sinG6#*&%&omegaGF*F)F*F*F/!\"\"\"\"!,$*&#F*\" \"#F**&,(F+F**(-%$cosGF-F*F)F*F/F*F0*()F)F5F*F+F*)F/F5F*F*F*F/!\"$F*F0 F*-%\"OG6#F*F5/F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(%\"tG\"\" \"-%$sinG6#*&%&omegaGF'F&F'F'F,!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Ret urn to example " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 86 "From the considerations in the previous s ubsection the general initial value problem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+2*lambda;" "6#,&*(%\"dG\" \"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F&F(%'lambdaGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+omega^2*x = F[0]*cos*omega*t;" "6#/,&*&%#d xG\"\"\"%#dtG!\"\"F'*&%&omegaG\"\"#%\"xGF'F'**&%\"FG6#\"\"!F'%$cosGF'F +F'%\"tGF'" }{TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt, t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "has the solution " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = F[0]/2;" "6#/- %\"xG6#%\"tG*&&%\"FG6#\"\"!\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "``(sin*om ega*t/(lambda*omega)-exp(-lambda*t)*sin(sqrt(omega^2-lambda^2)*t)/(lam bda*sqrt(omega^2-lambda^2)));" "6#-%!G6#,&**%$sinG\"\"\"%&omegaGF)%\"t GF)*&%'lambdaGF)F*F)!\"\"F)*(-%$expG6#,$*&F-F)F+F)F.F)-F(6#*&-%%sqrtG6 #,&*$F*\"\"#F)*$F-F=F.F)F+F)F)*&F-F)-F96#,&*$F*F=F)*$F-F=F.F)F.F." } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 31 "which has the approximat e form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = F[0 ]/2" "6#/-%\"xG6#%\"tG*&&%\"FG6#\"\"!\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "``(``(t*sin*omega*t/omega)-``((sin*omega*t-omega*t*cos*omega*t+omega^ 2*t^2*sin*omega*t)/(2*omega^3))*lambda )" "6#-%!G6#,&-F$6#*,%\"tG\"\" \"%$sinGF+%&omegaGF+F*F+F-!\"\"F+*&-F$6#*&,(*(F,F+F-F+F*F+F+*,F-F+F*F+ %$cosGF+F-F+F*F+F.*,F-\"\"#F*F8F,F+F-F+F*F+F+F+*&F8F+*$F-\"\"$F+F.F+%' lambdaGF+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 15 " is close to 0." }} {PARA 0 "" 0 "" {TEXT -1 39 "In the specific initial value problem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+1/60 \+ " "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(\"#gF,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt+9*x = 5*cos*3*t;" "6#/,&*&%#dxG \"\"\"%#dtG!\"\"F'*&\"\"*F'%\"xGF'F'**\"\"&F'%$cosGF'\"\"$F'%\"tGF'" } {TEXT -1 15 ", x(0) = 0, " }{XPPEDIT 18 0 "eval(dx/dt,t = 0) = 0;" "6#/-%%evalG6$*&%#dxG\"\"\"%#dtG!\"\"/%\"tG\"\"!F." }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 36 "considered in the first subsection, " } {XPPEDIT 18 0 "lambda=1/120" "6#/%'lambdaG*&\"\"\"F&\"$?\"!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "omega=3" "6#/%&omegaG\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F[0]=5" "6#/&%\"FG6#\"\"!\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "The approximate solution is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t) = 5*t*sin*3*t/6- (sin*3*t-3*t*cos*3*t+9*t^2*sin*3*t)/2592;" "6#/-%\"xG6#%\"tG,&*.\"\"& \"\"\"F'F+%$sinGF+\"\"$F+F'F+\"\"'!\"\"F+*&,(*(F,F+F-F+F'F+F+*,F-F+F'F +%$cosGF+F-F+F'F+F/*,\"\"*F+*$F'\"\"#F+F,F+F-F+F'F+F+F+\"%#f#F/F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Over the interval " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG " }{XPPEDIT 18 0 "``<5*Pi" "6#2%!G*&\"\"&\"\"\"%#PiGF'" }{TEXT -1 99 " , the graph of this approximate solution cannot be distinguished visua lly from the actual solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "h := t -> 100*sin(3*t)-3600 0/(19*sqrt(359))*exp(-t/120)*sin(19*sqrt(359)/120*t):\n'h(t)'=h(t);\nk := t->5/6*t*sin(3*t)-(sin(3*t)-3*t*cos(3*t)+9*t^2*sin(3*t))/2592:\n'k (t)'=k(t);\nplot([h(t),k(t)],t=0..5*Pi,color=[red,COLOR(RGB,.5,0,1)],t hickness=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"tG,&*& \"$+\"\"\"\"-%$sinG6#,$*&\"\"$F+F'F+F+F+F+*&#\"&+g$\"%@oF+*(\"$f$#F+\" \"#-%$expG6#,$*&\"$?\"!\"\"F'F+F@F+-F-6#,$**\"#>F+F?F@F7F8F'F+F+F+F+F@ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"kG6#%\"tG,**&#\"\"&\"\"'\"\" \"*&F'F--%$sinG6#,$*&\"\"$F-F'F-F-F-F-F-*&#F-\"%#f#F-F/F-!\"\"*&#F-\"$ k)F-*&-%$cosGF1F-F'F-F-F-*&#F-\"$)GF-*&)F'\"\"#F-F/F-F-F8" }}{PARA 13 "" 1 "" {GLPLOT2D 563 373 373 {PLOTDATA 2 "6&-%'CURVESG6%7gy7$$\"\"!F) F(7$$\"3mFK5j*))QU$!#=$\"3%*o/.2*zrV#F-7$$\"3<')Qy-FW8\\F-$\"3oa6!3ZsP 1%F-7$$\"3pWXYUk*HS'F-$\"3)R'f@!RkE*\\F-7$$\"3))oGU%='y@oF-$\"3KIQ@$p. 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#\"\"\"\"\"#F&**&%\"FG6#\"\"!F&%\"tGF&-%$sinG6#*&%&omegaGF&F-F&F&F2!\" \"F&F&*&#F&\"\"%F&**F)F&,(F.F&*(-%$cosGF0F&F2F&F-F&F3*()F-F'F&F.F&)F2F 'F&F&F&F2!\"$%'lambdaGF&F&F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"F G6#\"\"!\"\"&/%&omegaG\"\"$/%'lambdaG#\"\"\"\"$?\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**&#\"\"&\"\"'\"\"\"*&%\"tGF(-%$sinG6#,$*&\"\"$F(F*F (F(F(F(F(*&#F(\"%#f#F(F+F(!\"\"*&#F(\"$k)F(*&-%$cosGF-F(F*F(F(F(*&#F( \"$)GF(*&)F*\"\"#F(F+F(F(F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 35 "(a) Solve the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+4;" "6#,&*(% \"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(\"\"%F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dx/dt+4*x = 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "(b) Identify the transient and steady-state parts of the solution and plot their graphs along with the graph of the complete solution." }} {PARA 0 "" 0 "" {TEXT -1 47 "_________________________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________ " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 35 "(a) Solve the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+4*x = 5*exp(-t),x(0) = 0,eval (dx/dt,t = 0) = 0;" "6%/,&*(%\"dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\" \"F)*&\"\"%F)F(F)F)*&\"\"&F)-%$expG6#,$F,F-F)/-F(6#\"\"!F9/-%%evalG6$* &%#dxGF)%#dtGF-/F,F9F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "(b) Identify the transient and stea dy-state parts of the solution and plot their graphs along with the gr aph of the complete solution." }}{PARA 0 "" 0 "" {TEXT -1 47 "________ _______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "____________ ___________________________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 35 "(a) Solve the initial value p roblem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d* t^2)+225*x = 32*cos(17*t),x(0) = 0,eval(dx/dt,t = 0) = 0;" "6%/,&*(%\" dG\"\"#%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"$D#F)F(F)F)*&\"#KF)-%$co sG6#*&\"# " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "____________________ ___________________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 35 "(a) Solve the initial value problem" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+400 *x = 144*sin(16*t),x(0) = 0,eval(dx/dt,t = 0) = 36;" "6%/,&*(%\"dG\"\" #%\"xG\"\"\"*&F&F)*$%\"tGF'F)!\"\"F)*&\"$+%F)F(F)F)*&\"$W\"F)-%$sinG6# *&\"#;F)F,F)F)/-F(6#\"\"!F:/-%%evalG6$*&%#dxGF)%#dtGF-/F,F:\"#O" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 55 "and plot its graph over the interval from t = 0 to t = " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\" \"\"%#PiGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "(b) Show that the solution can be given in the form of a product of sine functions and replot the graph of the solut ion showing the envelope." }}{PARA 0 "" 0 "" {TEXT -1 47 "____________ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "____________________ ___________________________" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Code for dr awing picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 879 "PLOT(CURVES([[0,.5],[.05,.5],[.1,.8],[.2,.2],[. 3,.8],[.4,.2],\n[.5,.8],[.6,.2],[.7,.8],[.8,.2],[.9,.8],[1,.2],\n[1.1, .8],[1.2,.2],[1.3,.8],[1.4,.2],[1.5,.8],[1.6,.2],\n[1.7,.8],[1.8,.2],[ 1.9,.8],[2,.2],[2.1,.8],[2.2,.2],\n[2.3,.8],[2.4,.2],[2.5,.8],[2.6,.2] ,[2.7,.8],[2.8,.2],\n[2.9,.8],[3,.2],[3.05,.5],[3.1,.5]],\nTHICKNESS(2 ),COLOR(RGB,0,.6,0)),\nCURVES([[-.2,0],[5,0]],THICKNESS(2),COLOR(RGB,. 2,.2,.2)),\nCURVES([[-.3,1.1],[.2,1.1]],[[.1,1.15],[.2,1.1],[.1,1.05]] ,\n[[-.2,1.15],[-.3,1.1],[-.2,1.05]],THICKNESS(2),COLOR(RGB,.1,.1,.7)) ,\nPOLYGONS([[0,.01],[0,.99],[-.1,.99],[-.1,.01],[0,.01]],\nCOLOR(RGB, .6,.6,.6)),\nPOLYGONS([[3.1,.02],[3.1,.98],[3.6,.98],[3.6,.02],[3.1,.0 2]],\nCOLOR(RGB,.8,.8,.8)),\nPOLYGONS([[4.5,0],[4.5,1],[5,1],[5,0],[4. 5,0]],COLOR(RGB,.8,.5,0)),\nCURVES([[4,1],[5,1],[5,0]],[[3.6,.5],[4.5, .5]],[[4.5,0],[4.5,1]],\nTHICKNESS(2)),\nTEXT([3.35,.5],`M`),\nAXESSTY LE(NONE));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }