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"" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 326 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 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 "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 59 "Homogenous 2nd order linear DE's \+ with constant coefficients" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter St one, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 2 7.3.2007" }}{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 20 "Introductory example" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 86 "Second order differential equations arise in the description of many physical systems." }}{PARA 0 "" 0 "" {TEXT -1 36 "The simple har monic motion equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*t^2) = -omega^2*y;" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$ %\"tGF&F(!\"\",$*&%&omegaGF&F'F(F," }{TEXT -1 1 " " }}{PARA 258 "" 0 " " {TEXT -1 20 "is one such example." }}{PARA 0 "" 0 "" {TEXT -1 71 "We start by considering second order differential equations of the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a;" "6#%\"aG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(%\"dG\"\"#%\"y G\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF'%\"yGF'F'\" \"!" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+ " }{TEXT 278 1 "a" }{TEXT -1 2 ", " }{TEXT 279 1 "b" }{TEXT -1 5 " and " }{TEXT 280 1 "c" }{TEXT -1 15 " are constants." }}{PARA 0 "" 0 "" {TEXT -1 8 "It is a " }{TEXT 261 61 "homogeneous linear second order D E with constant coefficients" }{TEXT -1 142 ".\nThe meaning of the ter m \"constant coefficients\" is clear, but the meaning of other two ter ms \"linear\" and \"homogeneous\" will be given later." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the differential equation " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 0; " "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 13 " ------- (ii)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 13 "_ ____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Let us suppose that this equation has a solutio n of the form " }{XPPEDIT 18 0 "y(x) = exp(m*x);" "6#/-%\"yG6#%\"xG-%$ expG6#*&%\"mG\"\"\"F'F-" }{TEXT -1 5 ".\n " }}{PARA 0 "" 0 "" {TEXT -1 86 "We can check whether such a solution is possible by substitutin g it into the equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y(x) = exp(m*x);" "6#/-%\"yG 6#%\"xG-%$expG6#*&%\"mG\"\"\"F'F-" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([y*`'`(x) = m*exp (m*x),``],[y*`''`(x) = m^2*exp(m*x),``])" "6#-%*PIECEWISEG6$7$/*&%\"yG \"\"\"-%\"'G6#%\"xGF**&%\"mGF*-%$expG6#*&F0F*F.F*F*%!G7$/*&F)F*-%#''G6 #F.F**&F0\"\"#-F26#*&F0F*F.F*F*F5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "Substituting in the left hand side of (i) gives:" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y*`''`(x)+3*y*`'`(x) +2*y(x) = m^2*exp(m*x)+3*m*exp(m*x)+2*exp(m*x);" "6#/,(*&%\"yG\"\"\"-% #''G6#%\"xGF'F'*(\"\"$F'F&F'-%\"'G6#F+F'F'*&\"\"#F'-F&6#F+F'F',(*&%\"m GF2-%$expG6#*&F7F'F+F'F'F'*(F-F'F7F'-F96#*&F7F'F+F'F'F'*&F2F'-F96#*&F7 F'F+F'F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } {XPPEDIT 18 0 "y(x) = exp(m*x);" "6#/-%\"yG6#%\"xG-%$expG6#*&%\"mG\"\" \"F'F-" }{TEXT -1 40 " will be a solution of (ii) exactly when" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2*exp(m*x)+3*m*exp( m*x)+2*exp(m*x) = 0;" "6#/,(*&%\"mG\"\"#-%$expG6#*&F&\"\"\"%\"xGF,F,F, *(\"\"$F,F&F,-F)6#*&F&F,F-F,F,F,*&F'F,-F)6#*&F&F,F-F,F,F,\"\"!" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 "that is, when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(m*x)*(m^2+3*m+2) = 0" " 6#/*&-%$expG6#*&%\"mG\"\"\"%\"xGF*F*,(*$F)\"\"#F**&\"\"$F*F)F*F*F.F*F* \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "exp(m*x)<>0" "6#0-%$expG6#*&%\"mG\"\"\"%\"xGF)\"\"!" } {TEXT -1 31 " for all choices of the number " }{TEXT 291 1 "x" }{TEXT -1 37 ", this gives the quadratic equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+3*m+2=0" "6#/,(*$%\"mG\"\"#\"\"\"*& \"\"$F(F&F(F(F'F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "By factoring the left side of (ii) we can write this equation in t he form: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(m+2)* (m+1) = 0;" "6#/*&,&%\"mG\"\"\"\"\"#F'F',&F&F'F'F'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "which gives " }{XPPEDIT 18 0 "m \+ = -1" "6#/%\"mG,$\"\"\"!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "m = - 2" "6#/%\"mG,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus we see that " }{XPPEDIT 18 0 "y(x) = exp(-x);" "6#/-%\"yG6#% \"xG-%$expG6#,$F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(x) = exp (-2*x);" "6#/-%\"yG6#%\"xG-%$expG6#,$*&\"\"#\"\"\"F'F.!\"\"" }{TEXT -1 23 " are solutions of (ii)." }}{PARA 0 "" 0 "" {TEXT -1 69 "Because of the linearity of the differential operator it follows that" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp(-x)+ C[2]*exp(-2*x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#,$F'!\" \"F-F-*&&F+6#\"\"#F--F/6#,$*&F6F-F'F-F2F-F-" }{TEXT -1 13 " ------- (i i)" }}{PARA 0 "" 0 "" {TEXT -1 40 "is a solution of (ii) for any const ants " }{XPPEDIT 18 0 "C[1];" "6#&%\"CG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "It turns out that this is the " }{TEXT 261 16 "gener al solution" }{TEXT -1 133 " of the differential equation (ii); that i s, every possible solution of (i) must have the form (ii) for some cho ice of the constants " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "G iven additional information in the form of " }{TEXT 261 18 "initial co nditions" }{TEXT -1 9 " such as " }{XPPEDIT 18 0 "y(a)=b" "6#/-%\"yG6# %\"aG%\"bG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y*`'`(a)=c" "6#/*&%\"yG \"\"\"-%\"'G6#%\"aGF&%\"cG" }{TEXT -1 42 ", values can be assigned to \+ the constants " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 90 " so that \+ the given conditions are satisfied, leading to a specific solution, or so-called " }{TEXT 261 19 "particular solution" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "For examp le, we can find a solution curve which passes through the point " } {XPPEDIT 18 0 "``(0,1);" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 37 " with gra dient 3, that is, such that " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6# \"\"!\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = 3;" "6#/*& %\"yG\"\"\"-%\"'G6#\"\"!F&\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "y*`'`(x) = -C[1]*exp(-x)-2*C[2]*exp(-2*x);" "6#/*&%\"yG\"\"\"-% \"'G6#%\"xGF&,&*&&%\"CG6#F&F&-%$expG6#,$F*!\"\"F&F4*(\"\"#F&&F.6#F6F&- F16#,$*&F6F&F*F&F4F&F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 57 "Hence the initial conditions give rise to the equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1 = C[1]+C[2], ` `],[3 = -C[1]-2*C[2], ``]);" "6#-%*PIECEWISEG6$7$/\"\"\",&&%\"CG6#F(F( &F+6#\"\"#F(%!G7$/\"\"$,&&F+6#F(!\"\"*&F/F(&F+6#F/F(F7F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Adding these two equations gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4=-C[2]" "6#/\"\" %,$&%\"CG6#\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " } {XPPEDIT 18 0 "C[2] = -4" "6#/&%\"CG6#\"\"#,$\"\"%!\"\"" }{TEXT -1 20 ". Then the equation " }{XPPEDIT 18 0 "1 = C[1]+C[2];" "6#/\"\"\",&&% \"CG6#F$F$&F'6#\"\"#F$" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "C[1] = 5 ;" "6#/&%\"CG6#\"\"\"\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Thus the particular solution" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "y(x) = 5*exp(-x)-4*exp(-2*x);" "6#/-%\"yG6#%\"xG,& *&\"\"&\"\"\"-%$expG6#,$F'!\"\"F+F+*&\"\"%F+-F-6#,$*&\"\"#F+F'F+F0F+F0 " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 14 "__ ____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "satisfies the differential equation " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\" \"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx+2*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"! " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "subject to the initia l conditions " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = 3;" "6#/*&%\"yG\"\"\"-% \"'G6#\"\"!F&\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(5*exp(-x)-4*exp(-2*x), x=0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 467 231 231 {PLOTDATA 2 "6%-%'C URVESG6$7en7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a!#>$\"3?^FA@N&y9\"!#<7$ $\"3GLLL3x&)*3\"!#=$\"3II(HBeArE\"F27$$\"3))**\\i!R(*Rc\"F6$\"3EGT%f0+ 0N\"F27$$\"3umm\"H2P\"Q?F6$\"3EBI\\aZ;<9F27$$\"3!***\\PMnNrDF6$\"3%3E( *4)>du9F27$$\"3MLL$eRwX5$F6$\"3B1\">#>xu::F27$$\"3rLLL$eI8k$F6$\"38,Fn VD,V:F27$$\"33ML$3x%3yTF6$\"33=;\\ET,e:F27$$\"31nT5:j=XWF6$\"3Ox25G*e9 c\"F27$$\"3h+]PfyG7ZF6$\"3W.\\adw\\i:F27$$\"3gLek.%*Qz\\F6$\"3QZQ1oTJh :F27$$\"3emm\"z%4\\Y_F6$\"3#z()y@?\"3e:F27$$\"32++v$flu? Z:F27$$\"3`LLeR-/PiF6$\"3%=$G\"QN%yI:F27$$\"3]***\\il'pisF6$\"3e2$zid[ E[\"F27$$\"3>MLe*)>VB$)F6$\"3f'*fppO9=9F27$$\"3Y++DJbw!Q*F6$\"3,j;w9m= W8F27$$\"3%ommTIOo/\"F2$\"3MV.#f8)Hi7F27$$\"3YLL3_>jU6F2$\"39o6#H2.z= \"F27$$\"37++]i^Z]7F2$\"3_nmI^`\"Q5\"F27$$\"33++](=h(e8F2$\"35H\\*Q'>u ?5F27$$\"3/++]P[6j9F2$\"3n[$\\F2$\"3!>g&pJs, mhF67$$\"3K+]i!f#=$3#F2$\"3OK$*z&)4K1cF67$$\"3?+](=xpe=#F2$\"3twES=s#Q 6&F67$$\"37nm\"H28IH#F2$\"3k]I+2&e.k%F67$$\"3um;zpSS\"R#F2$\"38&)R1)yd ,C%F67$$\"3GLL3_?`(\\#F2$\"3A')=([.SN%QF67$$\"3fL$e*)>pxg#F2$\"3gi^Jb! ywY$F67$$\"33+]Pf4t.FF2$\"3=M<:$yT%oJF67$$\"3uLLe*Gst!GF2$\"3)emI]Z?C( GF67$$\"30+++DRW9HF2$\"3Wt$o3>bSf#F67$$\"3:++DJE>>IF2$\"3wQX,QahYBF67$ $\"3F+]i!RU07$F2$\"3Y:2'Qc^(G@F67$$\"3+++v=S2LKF2$\"3KviBegf4>F67$$\"3 Jmmm\"p)=MLF2$\"34tvWviNJ(>T?l:\"F67$$\"3O+]7.\"fF&QF2$\"3(fjwy!=/V5F67$$\"3Ymm ;/OgbRF2$\"3Hh]!))>4pU*F/7$$\"3w**\\ilAFjSF2$\"3uf_bi))4y%)F/7$$\"3yLL L$)*pp;%F2$\"3'\\6jcQ'[`wF/7$$\"3)RL$3xe,tUF2$\"3!f%y\"R!Q6#*oF/7$$\"3 Cn;HdO=yVF2$\"3/9%Hjf#36iF/7$$\"3a+++D>#[Z%F2$\"3>$p'zf>@WcF/7$$\"3Snm T&G!e&e%F2$\"3slY*4\"pJd]F/7$$\"3#RLLL)Qk%o%F2$\"3'>(z\"\\\"**)Qe%F/7$ $\"37+]iSjE!z%F2$\"3&4\\la!y\\FTF/7$$\"3a+]P40O\"*[F2$\"3fo\")e/5.LPF/ 7$$\"\"&F)$\"3YRxjFN\"3N$F/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABE LSG6$Q\"x6\"Q!F\\^l-%%VIEWG6$;F(F]]l%(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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 26 "Introductory example with " }{TEXT 0 6 "dsolve" }} {PARA 0 "" 0 "" {TEXT -1 25 "The general solution of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+3; " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\" F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "can be found as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "de := diff(y(x),x$2)+3*diff( y(x),x)+2*y(x)=0;\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"$F2-F(6$F* F-F2F2*&F1F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#% \"xG,&*&%$_C1G\"\"\"-%$expG6#,$F'!\"\"F+F+*&%$_C2GF+-F-6#,$F'!\"#F+F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The \+ particular solution which satisfies the initial conditions " } {XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "y*`'`(0) = 3;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&\"\"$ " }{TEXT -1 24 " can be found by giving " }{TEXT 0 6 "dsolve" }{TEXT -1 24 " this extra information." }}{PARA 0 "" 0 "" {TEXT -1 117 "The l ast statement in the group below constructs a function corresponding t o the solution, which can then be plotted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "de := diff(y(x),x $2)+3*diff(y(x),x)+2*y(x)=0;\nic := y(0)=1,D(y)(0)=3;\ndsolve(\{de,ic \},y(x));\nf := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"$F2-F(6$F*F -F2F2*&F1F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-% \"yG6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"yG6#%\"xG,&*&\"\"%\"\"\"-%$expG6#,$*&\"\"#F+F'F+!\"\"F+F2*&\"\" &F+-F-6#,$F'F2F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*&\"\"%\"\"\"-%$expG6#,$*&\"\"#F/9$F/!\"\"F /F7*&\"\"&F/-F16#,$F6F7F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(f(x),x=0..5,color=coral ,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 350 219 219 {PLOTDATA 2 " 6'-%'CURVESG6#7en7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a!#>$\"3?^FA@N&y9 \"!#<7$$\"3GLLL3x&)*3\"!#=$\"3II(HBeArE\"F27$$\"3))**\\i!R(*Rc\"F6$\"3 EGT%f0+0N\"F27$$\"3umm\"H2P\"Q?F6$\"3EBI\\aZ;<9F27$$\"3!***\\PMnNrDF6$ \"3%3E(*4)>du9F27$$\"3MLL$eRwX5$F6$\"3B1\">#>xu::F27$$\"3rLLL$eI8k$F6$ \"38,FnVD,V:F27$$\"33ML$3x%3yTF6$\"33=;\\ET,e:F27$$\"31nT5:j=XWF6$\"3O x25G*e9c\"F27$$\"3h+]PfyG7ZF6$\"3W.\\adw\\i:F27$$\"3gLek.%*Qz\\F6$\"3Q ZQ1oTJh:F27$$\"3emm\"z%4\\Y_F6$\"3#z()y@?\"3e:F27$$\"32++v$flu?Z:F27$$\"3`LLeR-/PiF6$\"3%=$G\"QN%yI:F27$$\"3]***\\il'pisF6$\"3 e2$zid[E[\"F27$$\"3>MLe*)>VB$)F6$\"3f'*fppO9=9F27$$\"3Y++DJbw!Q*F6$\"3 ,j;w9m=W8F27$$\"3%ommTIOo/\"F2$\"3MV.#f8)Hi7F27$$\"3YLL3_>jU6F2$\"39o6 #H2.z=\"F27$$\"37++]i^Z]7F2$\"3_nmI^`\"Q5\"F27$$\"33++](=h(e8F2$\"35H \\*Q'>u?5F27$$\"3/++]P[6j9F2$\"3n[$\\F2$\"3! >g&pJs,mhF67$$\"3K+]i!f#=$3#F2$\"3OK$*z&)4K1cF67$$\"3?+](=xpe=#F2$\"3t wES=s#Q6&F67$$\"37nm\"H28IH#F2$\"3k]I+2&e.k%F67$$\"3um;zpSS\"R#F2$\"38 &)R1)yd,C%F67$$\"3GLL3_?`(\\#F2$\"3A')=([.SN%QF67$$\"3fL$e*)>pxg#F2$\" 3gi^Jb!ywY$F67$$\"33+]Pf4t.FF2$\"3=M<:$yT%oJF67$$\"3uLLe*Gst!GF2$\"3)e mI]Z?C(GF67$$\"30+++DRW9HF2$\"3Wt$o3>bSf#F67$$\"3:++DJE>>IF2$\"3wQX,Qa hYBF67$$\"3F+]i!RU07$F2$\"3Y:2'Qc^(G@F67$$\"3+++v=S2LKF2$\"3KviBegf4>F 67$$\"3Jmmm\"p)=MLF2$\"34tvWviNJ(>T?l:\"F67$$\"3O+]7.\"fF&QF2$\"3(fjwy!=/V5F67$ $\"3Ymm;/OgbRF2$\"3Hh]!))>4pU*F/7$$\"3w**\\ilAFjSF2$\"3uf_bi))4y%)F/7$ $\"3yLLL$)*pp;%F2$\"3'\\6jcQ'[`wF/7$$\"3)RL$3xe,tUF2$\"3!f%y\"R!Q6#*oF /7$$\"3Cn;HdO=yVF2$\"3/9%Hjf#36iF/7$$\"3a+++D>#[Z%F2$\"3>$p'zf>@WcF/7$ $\"3SnmT&G!e&e%F2$\"3slY*4\"pJd]F/7$$\"3#RLLL)Qk%o%F2$\"3'>(z\"\\\"**) Qe%F/7$$\"37+]iSjE!z%F2$\"3&4\\la!y\\FTF/7$$\"3a+]P40O\"*[F2$\"3fo\")e /5.LPF/7$$\"\"&F)$\"3YRxjFN\"3N$F/-%+AXESLABELSG6$Q\"x6\"Q!Fe]l-%'COLO URG6&%$RGBG$\"*++++\"!\")$\")AR!)\\F]^lF(-%*THICKNESSG6#\"\"#-%%VIEWG6 $;F(F]]l%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Introductory example revisited" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Let D stand for t he differential operator " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dx G!\"\"" }{TEXT -1 9 " and let " }{XPPEDIT 18 0 "D^2;" "6#*$%\"DG\"\"# " }{TEXT -1 11 " stand for " }{XPPEDIT 18 0 "d^2/(d*x^2);" "6#*&%\"dG \"\"#*&F$\"\"\"*$%\"xGF%F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "Then the differential equation" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 0; " "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "can be written in the form" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(D^2+3*D+2)*y = 0; " "6#/*&,(*$%\"DG\"\"#\"\"\"*&\"\"$F)F'F)F)F(F)F)%\"yGF)\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "This equation can then be writ ten as" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(D+1)*(D+ 2)*y = 0;" "6#/*(,&%\"DG\"\"\"F'F'F',&F&F'\"\"#F'F'%\"yGF'\"\"!" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 29 "which is to be interpre ted as" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(D+1)*[(D+2 )*y] = 0;" "6#/*&,&%\"DG\"\"\"F'F'F'7#*&,&F&F'\"\"#F'F'%\"yGF'F'\"\"! " }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "The left side of this equation is the res ult of first applying the operator " }{XPPEDIT 18 0 "D+2;" "6#,&%\"DG \"\"\"\"\"#F%" }{TEXT -1 4 " to " }{TEXT 288 1 "y" }{TEXT -1 9 " to gi ve " }{XPPEDIT 18 0 "dy/dx+2*y;" "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*&\"\"# F&%\"yGF&F&" }{TEXT -1 20 ", and then applying " }{XPPEDIT 18 0 "D+1; " "6#,&%\"DG\"\"\"F%F%" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "dy/dx+2*y; " "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 9 " to g ive " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(D+1)*[dy/dx+ 2*y] = d/dx;" "6#/*&,&%\"DG\"\"\"F'F'F'7#,&*&%#dyGF'%#dxG!\"\"F'*&\"\" #F'%\"yGF'F'F'*&%\"dGF'F,F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[dy/dx+2* y]+[dy/dx+2*y] = d^2*y/(d*x^2)+3;" "6#/,&7#,&*&%#dyG\"\"\"%#dxG!\"\"F) *&\"\"#F)%\"yGF)F)F)7#,&*&F(F)F*F+F)*&F-F)F.F)F)F),&*(%\"dGF-F.F)*&F5F )*$%\"xGF-F)F+F)\"\"$F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y;" " 6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 87 "It follows that we can obtain a solution \+ for our second order differential equation in " }{TEXT 261 9 "two step s" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = (D+2)*y;" "6#/%\"uG*&,&%\"DG\"\" \"\"\"#F(F(%\"yGF(" }{TEXT -1 16 " in (i) to give " }{XPPEDIT 18 0 "(D +1)*u = 0;" "6#/*&,&%\"DG\"\"\"F'F'F'%\"uGF'\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 10 "first step" }{TEXT -1 26 " is to find a solution for" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(D+1)*u = 0;" "6#/*&,&%\"DG\"\"\"F'F'F'%\"uGF'\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "du/dx+u = 0;" "6#/,&*&%#duG\"\"\"%#dxG!\"\"F'%\"uG F'\"\"!" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 18 "This equatio n has " }{TEXT 256 1 "s" }{TEXT 261 18 "eparable variables" }{TEXT -1 22 " and can be written as" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(1/u,u) = -Int(1,x);" "6#/-%$IntG6$*&\"\"\"F(%\"uG! \"\"F),$-F%6$F(%\"xGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Thus we obtain the " }{TEXT 263 21 "intermediate solution" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x) = C[1]*exp(-x);" "6#/- %\"uG6#%\"xG*&&%\"CG6#\"\"\"F,-%$expG6#,$F'!\"\"F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 75 "A solution for the second order different ial equation must be a solution of" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "(D+2)*y = C[1]*exp(-x);" "6#/*&,&%\"DG\"\"\"\"\"#F'F '%\"yGF'*&&%\"CG6#F'F'-%$expG6#,$%\"xG!\"\"F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx+2*y = C[1]*exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\" \"F'*&\"\"#F'%\"yGF'F'*&&%\"CG6#F'F'-%$expG6#,$%\"xGF)F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 261 11 "second step" }{TEXT -1 23 " involves solving this \+ " }{TEXT 261 40 "first order linear differential equation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "A suitable integrating factor is \+ " }{XPPEDIT 18 0 "exp(Int(2,x)) = exp(2*x);" "6#/-%$expG6#-%$IntG6$\" \"#%\"xG-F%6#*&F*\"\"\"F+F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Multiplying by this integrating factor gives" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "exp(2*x);" "6#-%$expG6#*&\"\"#\"\"\"%\"xGF( " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*exp(2*x)*y = C[1]*exp(x);" " 6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"#F'-%$expG6#*&F+F'%\"xGF'F'%\"yGF' F'*&&%\"CG6#F'F'-F-6#F0F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[y*exp(2*x )] = C[1]*exp(x);" "6#/7#*&%\"yG\"\"\"-%$expG6#*&\"\"#F'%\"xGF'F'*&&% \"CG6#F'F'-F)6#F-F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "and so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*exp(2*x) = C [1]*exp(x)+C[2];" "6#/*&%\"yG\"\"\"-%$expG6#*&\"\"#F&%\"xGF&F&,&*&&%\" CG6#F&F&-F(6#F,F&F&&F06#F+F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "again giving the solution" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp(-x)+C[2]*exp(-2*x);" "6#/-%\"yG6#%\"xG ,&*&&%\"CG6#\"\"\"F--%$expG6#,$F'!\"\"F-F-*&&F+6#\"\"#F--F/6#,$*&F6F-F 'F-F2F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "for the seco nd order differential equation." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 72 "Solution of homogeneous 2nd order linear DE's with constant coefficients" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 60 "A homogeneous secon d order differential equation of the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d* x^2)+b;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = 0;" "6#/,&*&%#dyG\"\"\"%#d xG!\"\"F'*&%\"cGF'%\"yGF'F'\"\"!" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 275 1 "a" }{TEXT -1 2 ", " } {TEXT 276 1 "b" }{TEXT -1 5 " and " }{TEXT 277 1 "c" }{TEXT -1 89 " ar e constants, can be solved by the method outlined in the example of th e first section." }}{PARA 0 "" 0 "" {TEXT -1 35 "The first step is to \+ construct the " }{TEXT 261 18 "auxiliary equation" }}{PARA 256 "" 0 " " {TEXT -1 6 " " }{XPPEDIT 18 0 "a*m^2+b*m+c = 0;" "6#/,(*&%\"aG \"\"\"*$%\"mG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 13 " ---- --- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 42 "which is the equation to be s atisfied by " }{TEXT 281 1 "m" }{TEXT -1 17 " in order that " } {XPPEDIT 18 0 "y = exp(m*x);" "6#/%\"yG-%$expG6#*&%\"mG\"\"\"%\"xGF*" }{TEXT -1 21 "is a solution of (i)." }}{PARA 0 "" 0 "" {TEXT -1 30 "Th ere are 3 cases to consider." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 6 "Case I" }{TEXT -1 5 ": " }{XPPEDIT 18 0 "4*a *c < b^2;" "6#2*(\"\"%\"\"\"%\"aGF&%\"cGF&*$%\"bG\"\"#" }{TEXT 256 3 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "In this case the auxiliary quadrat ic equation (ii) has " }{TEXT 261 27 "two distinct real solutions" } {TEXT -1 7 ", say " }{XPPEDIT 18 0 "m[1];" "6#&%\"mG6#\"\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "m[2];" "6#&%\"mG6#\"\"#" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The general solution of (i) is:" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y = C[1]*exp(m[1]*x )+C[2]*exp(m[2]*x);" "6#/%\"yG,&*&&%\"CG6#\"\"\"F*-%$expG6#*&&%\"mG6#F *F*%\"xGF*F*F**&&F(6#\"\"#F*-F,6#*&&F06#F6F*F2F*F*F*" }{TEXT -1 2 " , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 17 "_________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 29 " and are arbitrary constants." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Example of case I" } {TEXT 259 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^ 2*y/(d*x^2)-dy/dx-6*y = 0;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xG F'F)!\"\"F)*&%#dyGF)%#dxGF-F-*&\"\"'F)F(F)F-\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 19 "Auxiliary equation:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "m^2-m-6 = 0;" "6#/,(*$%\"mG\"\"#\"\" \"F&!\"\"\"\"'F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "o r" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(m-3)*(m+2) = 0 ;" "6#/*&,&%\"mG\"\"\"\"\"$!\"\"F',&F&F'\"\"#F'F'\"\"!" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "giving" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "m = 3,m = -2;" "6$/%\"mG\"\"$/F$,$\"\"#!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "General solution of DE: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp (-2*x)+C[2]*exp(3*x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#, $*&\"\"#F-F'F-!\"\"F-F-*&&F+6#F3F--F/6#*&\"\"$F-F'F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "de := diff(y(x),x$2)-diff(y(x),x)-6*y(x)=0;\ndsolve(d e,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6 #%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-!\"\"*&\"\"'F2F*F2F5\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$expG6 #,$F'\"\"$F+F+*&%$_C2GF+-F-6#,$F'!\"#F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We can check this solution by s ubstituting it back into the differential equation." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(sol,d e);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*& %$_C1G\"\"\"-%$expG6#,$%\"xG!\"$F+F+*(%$_C2GF+F,F+F0F+F+-%\"$G6$F0\"\" #F+-F&6$F(F0!\"\"*(\"\"'F+F*F+F,F+F:**F " 0 "" {MPLTEXT 1 0 1 ";" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 7 "Case II" }{TEXT -1 5 ": \+ " }{XPPEDIT 18 0 "b^2 = 4*a*c;" "6#/*$%\"bG\"\"#*(\"\"%\"\"\"%\"aGF) %\"cGF)" }{TEXT 256 3 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "In this cas e the auxiliary quadratic equation has " }{TEXT 261 25 "exactly one re al solution" }{TEXT -1 7 ", say " }{XPPEDIT 18 0 "m = m[1];" "6#/%\"m G&F$6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "The gener al solution of (i) is:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "y = C[1]*exp(m[1]*x)+C[2]*x*exp(m[1]*x);" "6#/%\"yG,&*& &%\"CG6#\"\"\"F*-%$expG6#*&&%\"mG6#F*F*%\"xGF*F*F**(&F(6#\"\"#F*F2F*-F ,6#*&&F06#F*F*F2F*F*F*" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 16 "________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 25 " are arbitrary constants." }}{PARA 0 "" 0 "" {TEXT -1 150 "Certain ly we should expect there to be two arbitrary constants in the general solution from experience with other second order differential equatio ns." }}{PARA 0 "" 0 "" {TEXT -1 22 "The appearance of the " }{TEXT 261 7 "factor " }{TEXT 282 1 "x" }{TEXT -1 111 " in the second term ma y seem a bit mysterious at first sight. An explanation will be given i n the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "E xample of case II" }}{PARA 256 "" 0 "" {TEXT 260 0 "" }{TEXT -1 3 " \+ " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+6;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F( *$%\"xGF&F(!\"\"F(\"\"'F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+9*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"*F'%\"yGF'F'\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 19 "Auxiliary equation:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "m^2+6*m+9 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*& \"\"'F(F&F(F(\"\"*F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(m+3)^2 = 0 ;" "6#/*$,&%\"mG\"\"\"\"\"$F'\"\"#\"\"!" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 6 "giving" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "m = -3;" "6#/%\"mG,$\"\"$!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 23 "General solution of DE:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp(-3*x)+C[2]*x*exp (-3*x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"$F-F'F- !\"\"F-F-*(&F+6#\"\"#F-F'F--F/6#,$*&F3F-F'F-F4F-F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "de := diff(y(x),x$2)+6*diff(y(x),x)+9*y(x)=0;\nsol := dsolve(de, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#% \"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"'F2-F(6$F*F-F2F2*&\"\"*F2F*F2F2\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"yG6#%\"xG,&*&%$_C1G\"\"\" -%$expG6#,$F)!\"$F-F-*(%$_C2GF-F.F-F)F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We can check this solution by s ubstituting it back into the differential equation." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(sol,d e);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*& %$_C1G\"\"\"-%$expG6#,$%\"xG!\"$F+F+*(%$_C2GF+F,F+F0F+F+-%\"$G6$F0\"\" #F+*&\"\"'F+-F&6$F(F0F+F+*(\"\"*F+F*F+F,F+F+**F=F+F3F+F,F+F0F+F+\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Case III" }{TEXT -1 5 ": " } {XPPEDIT 18 0 "b^2 < 4*a*c;" "6#2*$%\"bG\"\"#*(\"\"%\"\"\"%\"aGF)%\"cG F)" }{TEXT 256 3 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "In this case the auxiliary quadratic equation has " }{TEXT 261 30 "two distinct comple x solutions" }{TEXT -1 13 " of the form " }{TEXT 293 1 "p" }{TEXT -1 1 " " }{TEXT 292 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q*i" "6#*&%\"qG \"\"\"%\"iGF%" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "q<>0" "6#0%\"qG\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "In fact the solutions are " }{XPPEDIT 18 0 "m = -b/(2* a)+``(sqrt(4*a*c-b^2)/(2*a))*i;" "6#/%\"mG,&*&%\"bG\"\"\"*&\"\"#F(%\"a GF(!\"\"F,*&-%!G6#*&-%%sqrtG6#,&*(\"\"%F(F+F(%\"cGF(F(*$F'F*F,F(*&F*F( F+F(F,F(%\"iGF(F(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "m = -b/(2*a)- ``(sqrt(4*a*c-b^2)/(2*a))*i;" "6#/%\"mG,&*&%\"bG\"\"\"*&\"\"#F(%\"aGF( !\"\"F,*&-%!G6#*&-%%sqrtG6#,&*(\"\"%F(F+F(%\"cGF(F(*$F'F*F,F(*&F*F(F+F (F,F(%\"iGF(F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "so " } {XPPEDIT 18 0 "p = -b/(2*a);" "6#/%\"pG,$*&%\"bG\"\"\"*&\"\"#F(%\"aGF( !\"\"F," }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q = sqrt(4*a*c-b^2)/(2* a);" "6#/%\"qG*&-%%sqrtG6#,&*(\"\"%\"\"\"%\"aGF,%\"cGF,F,*$%\"bG\"\"#! \"\"F,*&F1F,F-F,F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "The general solution of (i) is:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y = C[1]*exp(p*x)*sin*q*x +C[2]*exp(p*x)*cos*q*x;" "6#/%\"yG,&*,&%\"CG6#\"\"\"F*-%$expG6#*&%\"pG F*%\"xGF*F*%$sinGF*%\"qGF*F0F*F**,&F(6#\"\"#F*-F,6#*&F/F*F0F*F*%$cosGF *F2F*F0F*F*" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 267 24 "________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 25 " are arbitrary constants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Consider the special subcase of case III \+ where " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" } {TEXT -1 9 " we have " }{XPPEDIT 18 0 "p=0" "6#/%\"pG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " a*c>0" "6#2\"\"!*&%\"aG\"\"\"%\"cGF'" } {TEXT -1 10 ", so that " }{TEXT 320 1 "a" }{TEXT -1 5 " and " }{TEXT 321 1 "c" }{TEXT -1 21 " have the same sign. " }}{PARA 0 "" 0 "" {TEXT -1 77 "In this case the general solution of the second order dif ferential equation " }{TEXT 322 1 "a" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+ c*y =0" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F) !\"\"F)*&%\"cGF)F(F)F)\"\"!" }{TEXT -1 5 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=C[1]*cos*q*x+C[2]*sin*q*x" "6#/%\"yG, &**&%\"CG6#\"\"\"F*%$cosGF*%\"qGF*%\"xGF*F***&F(6#\"\"#F*%$sinGF*F,F*F -F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "q=sqrt(c/a" "6#/%\"qG-%%sqrtG6#*&%\"cG\"\"\"%\"aG!\"\" " }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Example of case III" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+6;" "6#,&*(%\"d G\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"'F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+13*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"#8F '%\"yGF'F'\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 19 "Auxiliary equation:" } }{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "m^2+6*m+13 = 0;" " 6#/,(*$%\"mG\"\"#\"\"\"*&\"\"'F(F&F(F(\"#8F(\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "(m+3)^2 = -4;" "6#/*$,&%\"mG\"\"\"\"\"$F'\"\"#,$\"\"%! \"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "giving" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "m = -3+2*i,m = -3-2*i;" "6 $/%\"mG,&\"\"$!\"\"*&\"\"#\"\"\"%\"iGF*F*/F$,&F&F'*&F)F*F+F*F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "General solution of DE: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp (-3*x)*sin(2*x)+C[2]*exp(-3*x)*cos(2*x);" "6#/-%\"yG6#%\"xG,&*(&%\"CG6 #\"\"\"F--%$expG6#,$*&\"\"$F-F'F-!\"\"F--%$sinG6#*&\"\"#F-F'F-F-F-*(&F +6#F9F--F/6#,$*&F3F-F'F-F4F--%$cosG6#*&F9F-F'F-F-F-" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "de := diff(y(x),x$2)+6*diff(y(x),x)+13*y(x)=0;\nsol := dsolve(de ,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6# %\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"'F2-F(6$F*F-F2F2*&\"#8F2F*F2F2\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"yG6#%\"xG,&*(%$_C1G\"\"\" -%$expG6#,$F)!\"$F--%$sinG6#,$F)\"\"#F-F-*(%$_C2GF-F.F--%$cosGF5F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We can check this solution by substituting it back into the differential equ ation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(sol,de);\nsimplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,*-%%diffG6$,&*(%$_C1G\"\"\"-%$expG6#,$%\"xG!\"$F+-%$s inG6#,$F0\"\"#F+F+*(%$_C2GF+F,F+-%$cosGF4F+F+-%\"$G6$F0F6F+*&\"\"'F+-F &6$F(F0F+F+**\"#8F+F*F+F,F+F2F+F+**FCF+F8F+F,F+F9F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "The c ase that the auxiliary equation has a single real solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 194 " To give an explanation for the form of the general solution in the cas e that the auxiliary equation has a single real solution we apply the \+ two stage operator method to the differential equation" }}{PARA 256 " " 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+6;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*& F%F(*$%\"xGF&F(!\"\"F(\"\"'F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+9 *y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"*F'%\"yGF'F'\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "Let D stand for the diff erential operator " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\" " }{TEXT -1 9 " and let " }{XPPEDIT 18 0 "D^2;" "6#*$%\"DG\"\"#" } {TEXT -1 11 " stand for " }{XPPEDIT 18 0 "d^2/(d*x^2);" "6#*&%\"dG\"\" #*&F$\"\"\"*$%\"xGF%F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "Then the equation can be written in the form" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "D^2+6*D+9*y = 0;" "6#/,(*$%\"DG\" \"#\"\"\"*&\"\"'F(F&F(F(*&\"\"*F(%\"yGF(F(\"\"!" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 36 "This equation can then be written as" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(D+3)*(D+3)*y = 0; " "6#/*(,&%\"DG\"\"\"\"\"$F'F',&F&F'F(F'F'%\"yGF'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "which is to be interpreted as" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(D+3)*[(D+3)*y] = 0; " "6#/*&,&%\"DG\"\"\"\"\"$F'F'7#*&,&F&F'F(F'F'%\"yGF'F'\"\"!" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 76 "The left side of this equation is the result of first a pplying the operator " }{XPPEDIT 18 0 "D+3;" "6#,&%\"DG\"\"\"\"\"$F%" }{TEXT -1 4 " to " }{TEXT 289 1 "y" }{TEXT -1 9 " to give " }{XPPEDIT 18 0 "dy/dx+3*y;" "6#,&*&%#dyG\"\"\"%#dxG!\"\"F&*&\"\"$F&%\"yGF&F&" } {TEXT -1 19 " and then applying " }{XPPEDIT 18 0 "D+3;" "6#,&%\"DG\"\" \"\"\"$F%" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "dy/dx+3*y;" "6#,&*&%#dyG \"\"\"%#dxG!\"\"F&*&\"\"$F&%\"yGF&F&" }{TEXT -1 9 " to give " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(D+3)*[dy/dx+3*y] = d/dx;" "6#/*&,&%\"DG\"\"\"\"\"$F'F'7#,&*&%#dyGF'%#dxG!\"\"F'*&F(F'%\"yGF'F'F' *&%\"dGF'F-F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "[dy/dx+3*y]+3*[dy/dx+3* y] = d^2*y/(d*x^2)+6;" "6#/,&7#,&*&%#dyG\"\"\"%#dxG!\"\"F)*&\"\"$F)%\" yGF)F)F)*&F-F)7#,&*&F(F)F*F+F)*&F-F)F.F)F)F)F),&*(%\"dG\"\"#F.F)*&F6F) *$%\"xGF7F)F+F)\"\"'F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+9*y;" "6 #,&*&%#dyG\"\"\"%#dxG!\"\"F&*&\"\"*F&%\"yGF&F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 87 "It follows that we can obtain a solution \+ for our second order differential equation in " }{TEXT 261 9 "two step s" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = (D+3)*y;" "6#/%\"uG*&,&%\"DG\"\" \"\"\"$F(F(%\"yGF(" }{TEXT -1 16 " in (i) to give " }{XPPEDIT 18 0 "(D +3)*u = 0;" "6#/*&,&%\"DG\"\"\"\"\"$F'F'%\"uGF'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 10 "first step" }{TEXT -1 27 " is to find a solution for" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(D+3)*u = 0;" "6#/*&,&%\"DG\"\"\"\"\"$F'F'%\"uGF'\" \"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "du/dx+3*u = 0;" "6#/,&*&%#duG\"\"\"%#dxG!\" \"F'*&\"\"$F'%\"uGF'F'\"\"!" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 18 "This equation has " }{TEXT 261 19 "separable variables" } {TEXT -1 22 " and can be written as" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Int(1/u,u) = -Int(3,x);" "6#/-%$IntG6$*&\"\"\"F(%\" uG!\"\"F),$-F%6$\"\"$%\"xGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "Thus we obtain the intermediate solution" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x) = C[1]*exp(-3*x);" "6#/-%\"uG6#%\" xG*&&%\"CG6#\"\"\"F,-%$expG6#,$*&\"\"$F,F'F,!\"\"F," }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 75 "A solution for the second order differen tial equation must be a solution of" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "(D+3)*y = C[1]*exp(-3*x);" "6#/*&,&%\"DG\"\"\"\"\"$ F'F'%\"yGF'*&&%\"CG6#F'F'-%$expG6#,$*&F(F'%\"xGF'!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx+3*y = C[1]*exp(-3*x);" "6#/,&*&%#dyG\"\"\"%# dxG!\"\"F'*&\"\"$F'%\"yGF'F'*&&%\"CG6#F'F'-%$expG6#,$*&F+F'%\"xGF'F)F' " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " The " }{TEXT 261 11 " second step" }{TEXT -1 23 " involves solving this " }{TEXT 261 40 "fir st order linear differential equation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "The integrating factor is " }{XPPEDIT 18 0 "exp(Int(3, x)) = exp(3*x);" "6#/-%$expG6#-%$IntG6$\"\"$%\"xG-F%6#*&F*\"\"\"F+F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Multiplying by this i ntegrating factor gives" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "exp(3*x); " "6#-%$expG6#*&\"\"$\"\"\"%\"xGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx+3*exp(3*x)*y = C[1];" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(\"\"$F'-%$e xpG6#*&F+F'%\"xGF'F'%\"yGF'F'&%\"CG6#F'" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[y*exp(3*x)] = C[1];" "6#/7#*&%\"yG\"\"\"-%$expG6#*&\"\"$F'%\"xGF'F '&%\"CG6#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "and so" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y*exp(3*x) = C[1]*x+C [2];" "6#/*&%\"yG\"\"\"-%$expG6#*&\"\"$F&%\"xGF&F&,&*&&%\"CG6#F&F&F,F& F&&F06#\"\"#F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "giving \+ the solution" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1]*x*exp(-3*x)+C[2]*exp(-3*x);" "6#/-%\"yG6#%\"xG,&*(&%\"CG6#\"\" \"F-F'F--%$expG6#,$*&\"\"$F-F'F-!\"\"F-F-*&&F+6#\"\"#F--F/6#,$*&F3F-F' F-F4F-F-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "The case that the auxiliary equation ha s complex solutions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 42 "As shown before, the differential equatio n" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d^2*y/(d*x^2)+6;" "6#,&*(%\"dG\" \"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"'F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+13*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"#8F '%\"yGF'F'\"\"!" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 26 "has the auxiliary equation" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " m^2+6*m+13 = 0;" "6#/,(*$%\"mG\"\"#\"\"\"*&\"\"'F(F&F(F(\"#8F(\"\"!" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "with the complex soluti ons " }{XPPEDIT 18 0 "m=-3" "6#/%\"mG,$\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 294 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*i" "6#*&\"\"#\"\"\"% \"iGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "This suggests \+ that the equation (i) has the solutions " }{XPPEDIT 18 0 "y = exp((-3+ 2*i)*x);" "6#/%\"yG-%$expG6#*&,&\"\"$!\"\"*&\"\"#\"\"\"%\"iGF.F.F.%\"x GF." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = exp((-3-2*i)*x);" "6#/%\" yG-%$expG6#*&,&\"\"$!\"\"*&\"\"#\"\"\"%\"iGF.F+F.%\"xGF." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "We can easily check this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "de := diff(y(x),x$2)+6*diff(y(x),x)+13*y(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*& \"\"'F2-F(6$F*F-F2F2*&\"#8F2F*F2F2\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sol := y(x) = exp((- 3+2*I)*x);\nsubs(sol,de);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%\"yG6#%\"xG-%$expG6#*&^$!\"$\"\"#\"\"\"F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%$expG6#*&^$!\"$\"\"#\"\"\"%\" xGF/-%\"$G6$F0F.F/*&\"\"'F/-F&6$F(F0F/F/*&\"#8F/F(F/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sol := y(x) = exp((-3-2*I)* x);\nsubs(sol,de);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $solG/-%\"yG6#%\"xG-%$expG6#*&^$!\"$!\"#\"\"\"F)F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(-%%diffG6$-%$expG6#*&^$!\"$!\"#\"\"\"%\"xGF/-%\"$G 6$F0\"\"#F/*&\"\"'F/-F&6$F(F0F/F/*&\"#8F/F(F/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 261 15 "Euler's fo rmula" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(i*theta) = cos*theta+i*sin* theta;" "6#/-%$expG6#*&%\"iG\"\"\"%&thetaGF),&*&%$cosGF)F*F)F)*(F(F)%$ sinGF)F*F)F)" }{TEXT -1 43 ", the two solutions above can be written a s" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "exp((-3+2*i)* x) = exp(-3*x)*(cos*2*x+i*sin*2*x);" "6#/-%$expG6#*&,&\"\"$!\"\"*&\"\" #\"\"\"%\"iGF-F-F-%\"xGF-*&-F%6#,$*&F)F-F/F-F*F-,&*(%$cosGF-F,F-F/F-F- **F.F-%$sinGF-F,F-F/F-F-F-" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "exp((-3-2*i)*x) = exp(-3*x)*(cos*2*x-i*sin*2*x);" "6#/- %$expG6#*&,&\"\"$!\"\"*&\"\"#\"\"\"%\"iGF-F*F-%\"xGF-*&-F%6#,$*&F)F-F/ F-F*F-,&*(%$cosGF-F,F-F/F-F-**F.F-%$sinGF-F,F-F/F-F*F-" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "Since each of these two functions is \+ a linear combination of the functions " }{XPPEDIT 18 0 "y = exp(-3*x)* sin*2*x;" "6#/%\"yG**-%$expG6#,$*&\"\"$\"\"\"%\"xGF,!\"\"F,%$sinGF,\" \"#F,F-F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = exp(-3*x)*cos*2*x; " "6#/%\"yG**-%$expG6#,$*&\"\"$\"\"\"%\"xGF,!\"\"F,%$cosGF,\"\"#F,F-F, " }{TEXT -1 146 ", albeit one involving the imaginary unit, we could s till try to see whether these two new functions can serve to generate \+ all possible solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sol := y(x)=exp(-3*x)*cos(2*x);\nsu bs(sol,de);\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/ -%\"yG6#%\"xG*&-%$expG6#,$F)!\"$\"\"\"-%$cosG6#,$F)\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$*&-%$expG6#,$%\"xG!\"$\"\"\"-%$ cosG6#,$F-\"\"#F/-%\"$G6$F-F4F/*&\"\"'F/-F&6$F(F-F/F/*(\"#8F/F)F/F0F/F /\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sol := y(x )=exp(-3*x)*sin(2*x);\nsubs(sol,de);\nsimplify(%); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$solG/-%\"yG6#%\"xG*&-%$expG6#,$F)!\"$\"\"\"-%$sinG 6#,$F)\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$*&-%$exp G6#,$%\"xG!\"$\"\"\"-%$sinG6#,$F-\"\"#F/-%\"$G6$F-F4F/*&\"\"'F/-F&6$F( F-F/F/*(\"#8F/F)F/F0F/F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\" !F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The general solution ca n be given in the form:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y(x) = C[1]*exp(-3*x)*cos*2*x+C[2]*exp(-3*x)*sin*2*x;" "6#/-%\"yG6#%\"xG,&*,&%\"CG6#\"\"\"F--%$expG6#,$*&\"\"$F-F'F-!\"\"F-%$ cosGF-\"\"#F-F'F-F-*,&F+6#F6F--F/6#,$*&F3F-F'F-F4F-%$sinGF-F6F-F'F-F- " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "In general, given" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a; " "6#%\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(% \"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+c*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF '%\"yGF'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "if th e auxiliary equation " }{XPPEDIT 18 0 "a*m^2+b*m+c = 0;" "6#/,(*&%\"a G\"\"\"*$%\"mG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 24 " has complex solutions " }{XPPEDIT 18 0 "m[1] = p+q*i;" "6#/&%\"mG6#\"\" \",&%\"pGF'*&%\"qGF'%\"iGF'F'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "m[ 2] = p-q*i;" "6#/&%\"mG6#\"\"#,&%\"pG\"\"\"*&%\"qGF*%\"iGF*!\"\"" } {TEXT -1 30 ", then the general solution is" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp(p*x)*cos*q*x+C[2]*exp (p*x)*sin*q*x;" "6#/-%\"yG6#%\"xG,&*,&%\"CG6#\"\"\"F--%$expG6#*&%\"pGF -F'F-F-%$cosGF-%\"qGF-F'F-F-*,&F+6#\"\"#F--F/6#*&F2F-F'F-F-%$sinGF-F4F -F'F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Summary" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 94 "Suppose we are give n a 2nd order homogeneous differential equation with constant coeffici ents:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F( *$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+c*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&%\"cGF'%\"yGF'F'\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "with auxiliary equation " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a*m^2+b*m+c = 0;" "6# /,(*&%\"aG\"\"\"*$%\"mG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT 262 6 "Case 1" }{TEXT 270 3 ": " } {XPPEDIT 18 0 "4*a*c < b^2" "6#2*(\"\"%\"\"\"%\"aGF&%\"cGF&*$%\"bG\"\" #" }{TEXT 285 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The auxiliary equation has " }{TEXT 261 27 "two distinct \+ real solutions" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m[1];" "6#&%\"mG6#\"\" \"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m[2];" "6#&%\"mG6#\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "General solution:" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y(x) = C[1]*exp(m[1] *x)+C[2]*exp(m[2]*x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F--%$expG6#* &&%\"mG6#F-F-F'F-F-F-*&&F+6#\"\"#F--F/6#*&&F36#F8F-F'F-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 6 "Case 2" }{TEXT 283 3 ": " }{XPPEDIT 18 0 "4*a*c = b^2" "6#/*( \"\"%\"\"\"%\"aGF&%\"cGF&*$%\"bG\"\"#" }{TEXT 286 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The auxiliary equati on has " }{TEXT 261 25 "exactly one real solution" }{TEXT -1 1 " " } {TEXT 290 1 "m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "General solution:" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y(x) \+ = C[1]*exp(m*x)+C[2]*x*exp(m*x);" "6#/-%\"yG6#%\"xG,&*&&%\"CG6#\"\"\"F --%$expG6#*&%\"mGF-F'F-F-F-*(&F+6#\"\"#F-F'F--F/6#*&F2F-F'F-F-F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 6 "Case 3" }{TEXT 284 3 ": " }{XPPEDIT 18 0 "b^2<4*a*c" "6# 2*$%\"bG\"\"#*(\"\"%\"\"\"%\"aGF)%\"cGF)" }{TEXT 287 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The auxiliary equ ation has " }{TEXT 261 18 "complex solutions " }{TEXT -1 1 " " } {XPPEDIT 18 0 "m[1] = p+q*i;" "6#/&%\"mG6#\"\"\",&%\"pGF'*&%\"qGF'%\"i GF'F'" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "m[2] = p-q*i;" "6#/&%\"mG6 #\"\"#,&%\"pG\"\"\"*&%\"qGF*%\"iGF*!\"\"" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "q<>0" "6#0%\"qG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 17 "General solution:" }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "y(x) = C[1]*exp(p*x)*cos*q*x+C[2]*exp(p*x)*sin*q*x; " "6#/-%\"yG6#%\"xG,&*,&%\"CG6#\"\"\"F--%$expG6#*&%\"pGF-F'F-F-%$cosGF -%\"qGF-F'F-F-*,&F+6#\"\"#F--F/6#*&F2F-F'F-F-%$sinGF-F4F-F'F-F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 297 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 61 "(a) Find the genera l solution of the differential equation 2 " }{XPPEDIT 18 0 "d^2*y/(d*x ^2)+9" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"*F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+10*y = 0;" "6#/,&*&%#dyG\"\"\"%#d xG!\"\"F'*&\"#5F'%\"yGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 103 "(b) Find the particular solution of the differential equ ation in (a) subject to the initial conditions " }{XPPEDIT 18 0 "y(0)= 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(0) = 4; " "6#/-%$y~'G6#\"\"!\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 100 "(c) Find the coordinates of the maximum point on the graph of \+ the particular solution found in (b). " }}{PARA 0 "" 0 "" {TEXT -1 60 "(d) Plot the graph of the particular solution found in (b). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary equation is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*m^2+9*m+10 = 0" "6#/,(*&\"\"#\"\"\"*$%\"mGF &F'F'*&\"\"*F'F)F'F'\"#5F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "Factoring the left hand side gives: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "(2*m+5)*(m+2)=0" "6#/*&,&*&\"\"#\"\" \"%\"mGF(F(\"\"&F(F(,&F)F(F'F(F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "m=-5/2" "6#/%\"mG,$*&\"\"&\" \"\"\"\"#!\"\"F*" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "m=-2" "6#/%\"mG ,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 176 "Since t he two solutions of the auxiliary equation are distinct real numbers, \+ the corresponding general solution of the given homogeneous second ord er differential equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = C[1]*exp(-5*x/2)+C[2]*exp(-2*x);" "6#/%\"yG,&*&&%\" CG6#\"\"\"F*-%$expG6#,$*(\"\"&F*%\"xGF*\"\"#!\"\"F3F*F**&&F(6#F2F*-F,6 #,$*&F2F*F1F*F3F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b ) " }}{PARA 0 "" 0 "" {TEXT -1 35 "The derivative of the solution is: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = `y '`(x); " "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$y~'G6#%\"xG" }{XPPEDIT 18 0 "``=-5/2" "6#/%!G,$*&\"\"&\"\"\"\"\"#!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C[ 1]*exp(-5*x/2)-2*C[2]*exp(-2*x)" "6#,&*&&%\"CG6#\"\"\"F(-%$expG6#,$*( \"\"&F(%\"xGF(\"\"#!\"\"F1F(F(*(F0F(&F&6#F0F(-F*6#,$*&F0F(F/F(F1F(F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 " x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\" \"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/dx = 4;" "6#/*&%#dyG\"\"\" %#dxG!\"\"\"\"%" }{TEXT -1 34 ", so we obtain the two equations: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 = C[1]+C[2]" "6#/\" \"!,&&%\"CG6#\"\"\"F)&F'6#\"\"#F)" }{TEXT -1 3 " , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4 = -5/2" "6#/\"\"%,$*&\"\"&\"\"\"\" \"#!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C[1]-2*C[2]" "6#,&&%\"CG6# \"\"\"F'*&\"\"#F'&F%6#F)F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "From the first equation w e have " }{XPPEDIT 18 0 "C[2]=-C[1]" "6#/&%\"CG6#\"\"#,$&F%6#\"\"\"!\" \"" }{TEXT -1 15 ", so replacing " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\" \"#" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "-C[1]" "6#,$&%\"CG6#\"\"\"!\" \"" }{TEXT -1 31 " in the second equation gives: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "4 = -5/2" "6#/\"\"%,$*&\"\"&\"\"\"\" \"#!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C[1]+2*C[1];" "6#,&&%\"CG6 #\"\"\"F'*&\"\"#F'&F%6#F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-1 /2;" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C[1] =4" "6#/&%\"CG6#\"\"\"\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "C[1] = -8;" "6#/&%\"CG6#\"\"\",$\"\")! \"\"" }{TEXT -1 11 ", and then " }{XPPEDIT 18 0 "C[2]=-C[1]" "6#/&%\"C G6#\"\"#,$&F%6#\"\"\"!\"\"" }{XPPEDIT 18 0 "`` = 8;" "6#/%!G\"\")" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The required particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 8*exp(-2*x)-8*exp(-5/2*x);" "6#/%\"yG,&*&\"\")\"\"\"-%$expG6#,$*&\"\" #F(%\"xGF(!\"\"F(F(*&F'F(-F*6#,$*(\"\"&F(F.F0F/F(F0F(F0" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 80 "Differentiating the expression for the particular solution, or substi tuting for " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 70 " in the expr ession for the derivative of the general solution, gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=-16*exp(-2*x)+20*exp( -5/2*x)" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&\"#;F&-%$expG6#,$*&\"\"#F&%\"x GF&F(F&F(*&\"#?F&-F-6#,$*(\"\"&F&F1F(F2F&F(F&F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 66 "The maximum point on the graph of the sol ution curve occurs where " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"% #dxG!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "This h appens when: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "16*e xp(-2*x)=20*exp(-5/2*x)" "6#/*&\"#;\"\"\"-%$expG6#,$*&\"\"#F&%\"xGF&! \"\"F&*&\"#?F&-F(6#,$*(\"\"&F&F,F.F-F&F.F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Multiplying bot h sides of this equation by " }{XPPEDIT 18 0 "1/16;" "6#*&\"\"\"F$\"#; !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(5/2*x)" "6#-%$expG6#*(\"\"& \"\"\"\"\"#!\"\"%\"xGF(" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(x/2)=5/4" "6#/-%$expG6#*&%\"xG\"\" \"\"\"#!\"\"*&\"\"&F)\"\"%F+" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "x /2 = ln(5/4);" "6#/*&%\"xG\"\"\"\"\"#!\"\"-%#lnG6#*&\"\"&F&\"\"%F(" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "x = 2*ln(5/4);" "6#/%\"xG*&\"\"#\"\"\"-%#lnG6#*&\"\"&F'\"\"%!\"\"F' " }{TEXT -1 1 " " }{TEXT 299 1 "~" }{TEXT -1 15 " 0.4462871026. " }} {PARA 0 "" 0 "" {TEXT -1 19 "The corresponding " }{TEXT 300 1 "y" } {TEXT -1 12 " value is: " }{XPPEDIT 18 0 "8*exp(4*ln(4/5))-8*exp(5*ln (4/5))=8*`.`*(4/5)^4-8*`.`*(4/5)^5" "6#/,&*&\"\")\"\"\"-%$expG6#*&\"\" %F'-%#lnG6#*&F,F'\"\"&!\"\"F'F'F'*&F&F'-F)6#*&F1F'-F.6#*&F,F'F1F2F'F'F 2,&*(F&F'%\".GF'*&F,F'F1F2F,F'*(F&F'F " 0 "" {MPLTEXT 1 0 40 "8*(4/5)^4-8*(4/5)^5;\nevalf(evalf(%,13));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"%[?\"%DJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++g`l!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "de := 2*diff(y(x),x$2)+9*di ff(y(x),x)+10*y(x)=0;\nic := y(0)=0,D(y)(0)=4;\ndsolve(\{de,ic\},y(x)) ;\ng := unapply(rhs(%),x):\n[solve(D(g)(x),x)];\nop(map(_x->[_x,g(_x)] ,%));\nevalf(evalf(%,13));\nplot(g(x),x=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"\"#\"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F0 F(F)F)*&\"\"*F)-F+6$F-F0F)F)*&\"#5F)F-F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\")\"\"\"-%$expG6#,$*&\" \"#F+F'F+!\"\"F+F+*&F*F+-F-6#,$*(\"\"&F+F1F2F'F+F2F+F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#,$*&\"\"#\"\"\"-%#lnG6##\"\"%\"\"&F'!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&\"\"#\"\"\"-%#lnG6##\"\"%\"\"&F' !\"\"#\"%[?\"%DJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+E5(GY%!#5$\" +++g`lF&" }}{PARA 13 "" 1 "" {GLPLOT2D 636 249 249 {PLOTDATA 2 "6%-%'C URVESG6$7do7$$\"\"!F)F(7$$\"3ILLL3x&)*3\"!#>$\"3G\"R(\\n]$QD%F-7$$\"3e mmm;arz@F-$\"34-`Z1Zh,$)F-7$$\"3')*****\\7t&pKF-$\"3U8Vg'\\&3:7!#=7$$ \"39LLLL3VfVF-$\"3#y\"eaBF)3e\"F:7$$\"3WmmmT&)G\\aF-$\"3x,\"fRdc#G>F:7 $$\"3s******\\i9RlF-$\"3%*y-J\"[()yD#F:7$$\"3+LLLeR+HwF-$\"3o/3(ffL/d# F:7$$\"3Hmmmm;')=()F-$\"3Ae;Fq9`mGF:7$$\"3KLLeR?ah5F:$\"3-&p2SF+WM$F:7 $$\"3-++]7z>^7F:$\"3!>Y71w7ux$F:7$$\"3qmmT&y`3W\"F:$\"3)['fvX?\\oTF:7$ $\"3RLLLe'40j\"F:$\"3\"z?'p'R%R?XF:7$$\"3rmm\"H_(zV=F:$\"3w\\#yj4pD([F :7$$\"3/++](Q&3d?F:$\"3'f*H2qy%>=&F:7$$\"3MLL3_KPqAF:$\"3!)R#=RUw(>%F:$\"3O\\)o'\\RcTlF:7$$\"3'*****\\(oZiH%F:$\"37mkdWf$*[l F:7$$\"3Wmmm;EI&R%F:$\"3Io*o_OWGb'F:7$$\"3!HLLeadV\\%F:$\"3W$R!*RLQMb' F:7$$\"3Q*****\\Z7Mf%F:$\"3AB!\\@1i3b'F:7$$\"3)GLLLLA:z%F:$\"3wZ\")f`> vOlF:7$$\"3Qmmm\">K'*)\\F:$\"3%[_Z/#\\d6lF:7$$\"3IKLLeZ*)*R&F:$\"35^u! [PZ%GkF:7$$\"3P*****\\Kd,\"eF:$\"38Ew2Az'*4jF:7$$\"3-mmm\"fX(emF:$\"3K -&)=f_?\")fF:7$$\"3.*****\\U7Y](F:$\"3?QIZJ\\pzbF:7$$\"3'QLLLV!pu$)F:$ \"3!H=sBp#F:7$$\"3?LLL`Q\"GT\"Fcv$\"3%)pgznq3-CF:7 $$\"3!*****\\s]k,:Fcv$\"3%*o')=\"f%='4#F:7$$\"39LLL`dF!e\"Fcv$\"3g$[^y NtG&=F:7$$\"33++]sgam;Fcv$\"3id-s.c#Rh\"F:7$$\"3/++]c#y( *fFcv$\"3wj D%40/N2\"F:7$$\"3immmTc-)*>Fcv$\"3++34'\\aMH*F-7$$\"3Mmm;f`@'3#Fcv$\"3 A&[>T\")*e')zF-7$$\"3y****\\nZ)H;#Fcv$\"3/2_RHv&F-7$$\"3f***** \\5a`T#Fcv$\"3K)**GshnjZ%F-7$$\"3o****\\7RV'\\#Fcv$\"3c2YE&zY2(QF-7$$ \"3k*****\\@fke#Fcv$\"3/*\\**)\\&Q-H$F-7$$\"3/LLL`4NnEFcv$\"3W)y)H]3rS GF-7$$\"3#*******\\,s`FFcv$\"3c=z&)pi>ECF-7$$\"3[mm;zM)>$GFcv$\"3fADhy q`,@F-7$$\"3$*******pfaG\"Q8\"F-7$$\"3#)****\\7yh]KFcv$\"3ky2)Rcezk*!#?7$$\"3xmmm')f dLLFcv$\"3#[q(4K5Ma#)F]_l7$$\"3bmmm,FT=MFcv$\"3k\">$ou-`LqF]_l7$$\"3FL L$e#pa-NFcv$\"3/^KN;eM)*fF]_l7$$\"3!*******Rv&)zNFcv$\"31#>R=(3%*z^F]_ l7$$\"3ILLLGUYoOFcv$\"3/lUq=9TwVF]_l7$$\"3_mmm1^rZPFcv$\"3)>)Qem!3Dw$F ]_l7$$\"34++]sI@KQFcv$\"3pR_YejK,KF]_l7$$\"34++]2%)38RFcv$\"3_<)pr5W=u #F]_l7$$\"\"%F)$\"3;;-7&e,0K#F]_l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AX ESLABELSG6$Q\"x6\"Q!Ffbl-%%VIEWG6$;F(Fgal%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 305 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 61 "(a) Find the general solution of the differential equation 4 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+4;" "6 #,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+37*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F' *&\"#PF'%\"yGF'F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 103 "(b) Find the particular solution of the differential equation in \+ (a) subject to the initial conditions " }{XPPEDIT 18 0 "y(0)=0" "6#/-% \"yG6#\"\"!F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(0) = 3;" "6#/-%$ y~'G6#\"\"!\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "(c) \+ Find the " }{TEXT 307 1 "x" }{TEXT -1 53 " intercepts of the particula r solution found in (b). " }}{PARA 0 "" 0 "" {TEXT -1 153 "(d) Find th e coordinates of the first turning point on the graph of the particula r solution found in (b), that is, the turning point to the right of th e " }{TEXT 316 1 "y" }{TEXT -1 19 " axis which has an " }{TEXT 315 1 " x" }{TEXT -1 32 " coordinate of least magnitude. " }}{PARA 0 "" 0 "" {TEXT -1 60 "(e) Plot the graph of the particular solution found in (b ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "So lution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary equation is:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "4*m^2+4*m+37 = 0;" "6#/,(*&\"\"%\"\" \"*$%\"mG\"\"#F'F'*&F&F'F)F'F'\"#PF'\"\"!" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Comparing with the standard equation " }{XPPEDIT 18 0 "a*m^2+b*m+c=0" "6#/,(*&%\"aG\"\" \"*$%\"mG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "b^2<4*a*c" "6#2*$%\"bG\"\"#*(\"\"%\"\"\"%\"aGF)%\"c GF)" }{TEXT -1 29 ", so the roots are complex. " }}{PARA 0 "" 0 "" {TEXT -1 42 "The equation can be written in the form: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*m^2+4*m=-37" "6#/,&*&\"\"%\"\" \"*$%\"mG\"\"#F'F'*&F&F'F)F'F',$\"#P!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "Dividing both sides of the equation by 4 gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+m = -37/4;" " 6#/,&*$%\"mG\"\"#\"\"\"F&F(,$*&\"#PF(\"\"%!\"\"F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 7 "Adding " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F $\"\"%!\"\"" }{TEXT -1 64 " to both sides \"completes the square\" on \+ the left side to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m^2+m+1/4=-36/4" "6#/,(*$%\"mG\"\"#\"\"\"F&F(*&F(F(\"\"%!\"\"F(, $*&\"#OF(F*F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(m+1/2)^2=-9" "6#/*$, &%\"mG\"\"\"*&F'F'\"\"#!\"\"F'F),$\"\"*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "m+1/2 =`` " "6#/,&%\"mG\"\"\"*&F&F&\"\"#!\"\"F&%!G" } {TEXT 309 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3*i" "6#*&\"\"$\"\"\"% \"iGF%" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m= -1/2" "6#/%\"mG, $*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{TEXT 308 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3*i" "6#*&\"\"$\"\"\"%\"iGF%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The corre sponding general solution of the given homogeneous second order differ ential equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=C[1]*exp(-x/2)*cos*3*x+C[2]*exp(-x/2)*sin*3*x" "6#/%\"yG,&*,&%\"C G6#\"\"\"F*-%$expG6#,$*&%\"xGF*\"\"#!\"\"F2F*%$cosGF*\"\"$F*F0F*F**,&F (6#F1F*-F,6#,$*&F0F*F1F2F2F*%$sinGF*F4F*F0F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 5 "When \+ " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 4 " so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 = C[1]*exp(0)*cos*0+0;" "6#/\"\"!,&**&%\"CG 6#\"\"\"F*-%$expG6#F$F*%$cosGF*F$F*F*F$F*" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "C[1]=0" "6#/&%\"CG6#\"\" \"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = C[2]*exp(-x/2 )*sin*3*x" "6#/%\"yG*,&%\"CG6#\"\"#\"\"\"-%$expG6#,$*&%\"xGF*F)!\"\"F1 F*%$sinGF*\"\"$F*F0F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 94 "We can obtain the derivative of the s olution by using the rule for differentiating a product: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = `y '`(x);" "6#/*&%# dyG\"\"\"%#dxG!\"\"-%$y~'G6#%\"xG" }{XPPEDIT 18 0 "``=C[2]*(``(-1/2)*` .`*exp(-x/2)*sin*3*x+exp(-x/2)*`.`*3*cos*3*x)" "6#/%!G*&&%\"CG6#\"\"# \"\"\",&*.-F$6#,$*&F*F*F)!\"\"F1F*%\".GF*-%$expG6#,$*&%\"xGF*F)F1F1F*% $sinGF*\"\"$F*F8F*F**.-F46#,$*&F8F*F)F1F1F*F2F*F:F*%$cosGF*F:F*F8F*F*F *" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "dy/dx = 3;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"$" }{TEXT -1 28 ", so we obtain the equ ation " }{XPPEDIT 18 0 "3 = 3*C[2];" "6#/\"\"$*&F$\"\"\"&%\"CG6#\"\"#F &" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "C[2]=1" "6#/&%\"CG6# \"\"#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "The required particular solution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=exp(-x/2)*sin*3*x" "6#/% \"yG**-%$expG6#,$*&%\"xG\"\"\"\"\"#!\"\"F.F,%$sinGF,\"\"$F,F+F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 310 1 "x" }{TEXT -1 46 " intercepts of the so lution curve occur where " }{XPPEDIT 18 0 "exp(-x/2)*sin*3*x=0" "6#/** -%$expG6#,$*&%\"xG\"\"\"\"\"#!\"\"F-F+%$sinGF+\"\"$F+F*F+\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "This only happens when \+ " }{XPPEDIT 18 0 "sin*3*x=0" "6#/*(%$sinG\"\"\"\"\"$F&%\"xGF&\"\"!" } {TEXT -1 13 " which gives " }{XPPEDIT 18 0 "3*x=k*Pi" "6#/*&\"\"$\"\" \"%\"xGF&*&%\"kGF&%#PiGF&" }{TEXT -1 8 ", where " }{TEXT 311 1 "k" } {TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 312 1 "x" }{TEXT -1 25 " intercepts are given by " }{XPPEDIT 18 0 "x=k*Pi/3" "6#/%\"xG*(%\"kG\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 8 ", w here " }{TEXT 313 1 "k" }{TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 314 1 "x" }{TEXT -1 55 " intercepts to t he right of the origin are therefore: " }{XPPEDIT 18 0 "Pi/3,2*Pi/3,P i,4*Pi/3,5*Pi/3,2*Pi,7*Pi/3,` . . . `" "6**&%#PiG\"\"\"\"\"$!\"\"*(\" \"#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'%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 80 "Diffe rentiating the expression for the particular solution, or substituting for " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 70 " in the e xpression for the derivative of the general solution, gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1/2" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&F&F&\"\"#F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x /2)*(6*cos*3*x-sin*3*x)" "6#*&-%$expG6#,$*&%\"xG\"\"\"\"\"#!\"\"F,F*,& **\"\"'F*%$cosGF*\"\"$F*F)F*F**(%$sinGF*F1F*F)F*F,F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 63 "Turning points on the graph of the so lution curve occurs where " }{XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\" %#dxG!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "This \+ happens when: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "6*c os*3*x=sin*3*x" "6#/**\"\"'\"\"\"%$cosGF&\"\"$F&%\"xGF&*(%$sinGF&F(F&F )F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan*3*x=6" "6#/*(%$ta nG\"\"\"\"\"$F&%\"xGF&\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "The first positive solution of this equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=1/3" "6#/%\"xG*&\"\"\"F&\"\"$ !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*6" "6#*&%'arctanG\"\"\" \"\"'F%" }{TEXT -1 1 " " }{TEXT 317 1 "~" }{TEXT -1 15 " 0.4685492165. " }}{PARA 0 "" 0 "" {TEXT -1 19 "The corresponding " }{TEXT 318 1 "y " }{TEXT -1 16 " coordinate is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = exp(arctan*6/6)*sin(arctan(6));" "6#/%\"yG*&-%$expG 6#*(%'arctanG\"\"\"\"\"'F+F,!\"\"F+-%$sinG6#-F*6#F,F+" }{XPPEDIT 18 0 "``=6/sqrt(37)" "6#/%!G*&\"\"'\"\"\"-%%sqrtG6#\"#P!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(arctan*6/6);" "6#-%$expG6#*(%'arctanG\"\"\"\"\" 'F(F)!\"\"" }{TEXT -1 1 " " }{TEXT 319 1 "~" }{TEXT -1 14 " 1.24679358 2. " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 50 ": This turning point is actually a maximum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "de := 4*diff(y(x),x$2)+4* diff(y(x),x)+37*y(x)=0;\nic := y(0)=0,D(y)(0)=3;\ndsolve(\{de,ic\},y(x ));\ng := unapply(rhs(%),x):\n[solve(D(g)(x),x)];\nop(map(_x->[_x,g(_x )],%));\nevalf(evalf(%,13));\nplot(g(x),x=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"\"%\"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F0 \"\"#F)F)*&F(F)-F+6$F-F0F)F)*&\"#PF)F-F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&-%$expG6#,$*&\"\"#!\"\"F'\" \"\"F/F0-%$sinG6#,$*&\"\"$F0F'F0F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7#,$*&#\"\"\"\"\"$F'-%'arctanG6#\"\"'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*&#\"\"\"\"\"$F'-%'arctanG6#\"\"'F'F',$*&#F,\"#PF'* &-%$expG6#,$*&#F'F,F'F)F'!\"\"F'F0#F'\"\"#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+l@\\&o%!#5$\"+&e,Q!yF&" }}{PARA 13 "" 1 "" {GLPLOT2D 636 249 249 {PLOTDATA 2 "6%-%'CURVESG6$7ep7$$\"\"!F)F(7$$\"3 emmm;arz@!#>$\"3Q^dQJqljkF-7$$\"39LLLL3VfVF-$\"3ad23:g)fF\"!#=7$$\"3s* *****\\i9RlF-$\"3Ig(*\\ge[')=F57$$\"3Hmmmm;')=()F-$\"3!)QiAlWivCF57$$ \"3-++]7z>^7F5$\"3BTPL()ouVMF57$$\"3RLLLe'40j\"F5$\"3d(HIxb64L%F57$$\" 3/++](Q&3d?F5$\"3uj'GRy48A&F57$$\"3mmmm;6m$[#F5$\"3A0!y8Z3'))fF57$$\"3 jmmmmW18HF5$\"3v\"z8u#[4HmF57$$\"3fmmm;yYULF5$\"3!)\\Ez6j9KrF57$$\"3/+ +](GI)pPF5$\"3#zif/@gP\\(F57$$\"3%HLL$eF>(>%F5$\"3wEGTTp[;xF57$$\"3'** ***\\(oZiH%F5$\"3UJJy>cY[xF57$$\"3Wmmm;EI&R%F5$\"3s9q#Q%=8txF57$$\"3!H LLeadV\\%F5$\"3E6ug9a`!z(F57$$\"3Q*****\\Z7Mf%F5$\"3*3%)e3?K2!yF57$$\" 3'emmTSnCp%F5$\"3EW&G(GSy.yF57$$\"3)GLLLLA:z%F5$\"3ExXAn&e(*z(F57$$\"3 !*****\\isd!*[F5$\"3wsl^4#H()y(F57$$\"3Qmmm\">K'*)\\F5$\"3q%HP-&\\xqxF 57$$\"3IKLLeZ*)*R&F5$\"3\"G(ozk!3Yi(F57$$\"3P*****\\Kd,\"eF5$\"3gNH)Ri g\"otF57$$\"3-mmm\"fX(emF5$\"3HlP0)*=0DlF57$$\"3_KLL3!z;3(F5$\"3[>CwPI ^pfF57$$\"3.*****\\U7Y](F5$\"3&\\p'zHFSS`F57$$\"3Wmm;H9lRzF5$\"3c'[CXD e/j%F57$$\"3'QLLLV!pu$)F5$\"3usN?'pF:(QF57$$\"3K+++DI(yv)F5$\"3WeYn62Z uJF57$$\"3xmmm;c0T\"*F5$\"3QC,5(Gn8Y#F57$$\"3+LLLe%GCd*F5$\"3Q%[d'H_&> l\"F57$$\"3#*******H,Q+5!#<$\"3ua&*y#*fE*[)F-7$$\"3&*******RXpV5F_u$\" 3R5HXK3>Oi!#?7$$\"3)*******\\*3q3\"F_u$!3s5f.<7=>pF-7$$\"3()******4/vG 6F_u$!3wQ+Tn7dx8F57$$\"3)*******p=\\q6F_u$!3#Hd*4zvZ8?F57$$\"3@LLe9rR3 7F_u$!3McUAg\"36a#F57$$\"3mmm;fBIY7F_u$!3SDrb9$4g,$F57$$\"3GLLLj$[kL\" F_u$!3()GcRm\"\\1\"RF57$$\"3MLLL36ju8F_u$!3Yh01'R2K=%F57$$\"3?LLL`Q\"G T\"F_u$!3S#zf!)em-R%F57$$\"3$***\\7e;-N9F_u$!3Q3m^S5U![%F57$$\"3mmm\"H YHsX\"F_u$!3mpp%o#))Q[XF57$$\"3=L$3xEP%z9F_u$!3I@g:+2N%f%F57$$\"3!**** *\\s]k,:F_u$!3R:G&)*o#e=YF57$$\"3@L$3Fu-8_\"F_u$!3t\\c*z/&>AYF57$$\"3_ mm\"HTg4a\"F_u$!3G[:+;;Q4YF57$$\"3%)**\\7$3=1c\"F_u$!3GQRatu]!e%F57$$ \"39LLL`dF!e\"F_u$!3FR<,Cu)f`%F57$$\"3^mm\"H\"4TB;F_u$!3iKKkY?z&Q%F57$ $\"33++]sgam;F_u$!3i^LhoZ;oTF57$$\"3/++]F_u$!3C'*=D;*Hz)>F57$$\"3 immmTc-)*>F_u$!3AX(e^z^)\\5F57$$\"3Mmm;f`@'3#F_u$!3c'[Q$>_sX')Fgu7$$\" 3y****\\nZ)H;#F_u$\"3.clr'=o#GpF-7$$\"3YmmmJy*eC#F_u$\"3W!p9#Gs@G9F57$ $\"3')******R^bJBF_u$\"3i:#o*eJ8N?F57$$\"3f*****\\5a`T#F_u$\"3q\\(p-q4 MX#F57$$\"3k***\\(3S*eX#F_u$\"3#[c@(*ew$*e#F57$$\"3o****\\7RV'\\#F_u$ \"3c.:CFMT\"o#F57$$\"3X**\\7Q-%*=DF_u$\"3&onQ5!pl8FF57$$\"3m***\\PcY9a #F_u$\"30^utE]hKFF57$$\"3))**\\P*)G&Rc#F_u$\"3hAF@oc\\QFF57$$\"3k***** \\@fke#F_u$\"37!e]\"HEcJFF57$$\"3Nmm;%30pi#F_u$\"3y/%)\\s,%zo#F57$$\"3 /LLL`4NnEF_u$\"3gHQ6H`B1EF57$$\"3Emmm^b`5FF_u$\"3'\\%RIL$Q+[#F57$$\"3# *******\\,s`FF_u$\"3nx#)\\vTO$GF_u$\"32])*fMuqV>F57$ $\"3$*******pfay'\\z$F-7$$\"3;LLL$)G[kJF_u$!3oGltD %p,T\"F-7$$\"3#)****\\7yh]KF_u$!3DUH`=WKCjF-7$$\"3xmmm')fdLLF_u$!3d5d5 a\\9%G(f\"F57$$\"3\"*****\\n'*33PF_u$! 3c5*H$3R2`:F57$$\"3_mmm1^rZPF_u$!3%p1*>c?\\)[\"F57$$\"34++]sI@KQF_u$!3 ')[+7dS'3H\"F57$$\"34++]2%)38RF_u$!34'Gyj@r-/\"F57$$\"\"%F)$!3Ii%oMyC< E(F--%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F[hl-%%VI EWG6$;F(F\\gl%(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 "" }}{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 295 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 61 "(a) Find the general solution of the differential \+ equation 9 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+12;" "6#,&*(%\"dG\"\"#%\"yG \"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"#7F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 " dy/dx+4*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F'\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 103 "(b) Find the partic ular solution of the differential equation in (a) subject to the initi al conditions " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(0) = -2;" "6#/-%$y~'G6#\"\"!,$\" \"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "(c) Find the " }{TEXT 301 1 "x" }{TEXT -1 52 " intercept of the particular solutio n found in (b). " }}{PARA 0 "" 0 "" {TEXT -1 100 "(d) Find the coordin ates of the minimum point on the graph of the particular solution foun d in (b). " }}{PARA 0 "" 0 "" {TEXT -1 60 "(e) Plot the graph of the p articular solution found in (b). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 296 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 26 "The auxiliary eq uation is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "9*m^2+1 2*m+4 = 0;" "6#/,(*&\"\"*\"\"\"*$%\"mG\"\"#F'F'*&\"#7F'F)F'F'\"\"%F'\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 37 "Comparing with the standard equation " }{XPPEDIT 18 0 " a*m^2+b*m+c=0" "6#/,(*&%\"aG\"\"\"*$%\"mG\"\"#F'F'*&%\"bGF'F)F'F'%\"cG F'\"\"!" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "b^2 = 4*a*c;" "6#/* $%\"bG\"\"#*(\"\"%\"\"\"%\"aGF)%\"cGF)" }{TEXT -1 56 ", so the auxilia ry equation has exactly one real root. " }}{PARA 0 "" 0 "" {TEXT -1 41 "The equation can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(3*m+2)^2 = 0;" "6#/*$,&*&\"\"$\"\"\"% \"mGF(F(\"\"#F(F*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "m = -2/3;" "6#/%\"mG,$*& \"\"#\"\"\"\"\"$!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "The corresponding general solution of the given homogeneous second order differential equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = C[1]*exp(-2*x/3)+C[2]*x*exp(-2*x/3);" "6#/%\"yG,&* &&%\"CG6#\"\"\"F*-%$expG6#,$*(\"\"#F*%\"xGF*\"\"$!\"\"F3F*F**(&F(6#F0F *F1F*-F,6#,$*(F0F*F1F*F2F3F3F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 94 "We can obtain the der ivative of the solution by using the rule for differentiating a produc t: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = `y '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$y~' G6#%\"xG" }{XPPEDIT 18 0 "`` = -2/3;" "6#/%!G,$*&\"\"#\"\"\"\"\"$!\"\" F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C[1]*exp(-2*x/3)+C[2]*exp(-2*x/3) \+ +C[2]*x*``(-2/3)*exp(-2*x/3)" "6#,(*&&%\"CG6#\"\"\"F(-%$expG6#,$*(\" \"#F(%\"xGF(\"\"$!\"\"F1F(F(*&&F&6#F.F(-F*6#,$*(F.F(F/F(F0F1F1F(F(**&F &6#F.F(F/F(-%!G6#,$*&F.F(F0F1F1F(-F*6#,$*(F.F(F/F(F0F1F1F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=0" "6 #/%\"xG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/dx = -2;" "6#/*&%#dyG\"\"\"%# dxG!\"\",$\"\"#F(" }{TEXT -1 34 ", so we obtain the two equations: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1=C[1]*exp(0)+0" "6# /\"\"\",&*&&%\"CG6#F$F$-%$expG6#\"\"!F$F$F-F$" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-2=-2/3" "6#/,$\"\"#! \"\",$*&F%\"\"\"\"\"$F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C[1]+C[2]" "6#,&&%\"CG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The first equation gives " }{XPPEDIT 18 0 "C[1]=1" "6#/&% \"CG6#\"\"\"F'" }{TEXT -1 11 ", and then " }{XPPEDIT 18 0 "C[2]=-2+2/3 " "6#/&%\"CG6#\"\"#,&F'!\"\"*&F'\"\"\"\"\"$F)F+" }{XPPEDIT 18 0 "``=-4 /3" "6#/%!G,$*&\"\"%\"\"\"\"\"$!\"\"F*" }{TEXT -1 27 " from the second equation. " }}{PARA 0 "" 0 "" {TEXT -1 37 "The required particular so lution is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=exp( -2*x/3)-4/3" "6#/%\"yG,&-%$expG6#,$*(\"\"#\"\"\"%\"xGF,\"\"$!\"\"F/F,* &\"\"%F,F.F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*exp(-2*x/3)" "6#*&% \"xG\"\"\"-%$expG6#,$*(\"\"#F%F$F%\"\"$!\"\"F-F%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=exp(-2*x/3)*(1-4*x/3)" "6#/%\"yG*&-%$expG6#,$*(\"\"# \"\"\"%\"xGF,\"\"$!\"\"F/F,,&F,F,*(\"\"%F,F-F,F.F/F/F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 302 1 "x" }{TEXT -1 46 " intercepts of the solution curve \+ occur where " }{XPPEDIT 18 0 "exp(-2*x/3)*(1-4*x/3)=0" "6#/*&-%$expG6# ,$*(\"\"#\"\"\"%\"xGF+\"\"$!\"\"F.F+,&F+F+*(\"\"%F+F,F+F-F.F.F+\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Hence there is a singl e " }{TEXT 303 1 "x" }{TEXT -1 30 " intercept which occurs where " } {XPPEDIT 18 0 "x=3/4" "6#/%\"xG*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) \+ " }}{PARA 0 "" 0 "" {TEXT -1 80 "Differentiating the expression for th e particular solution, or substituting for " }{XPPEDIT 18 0 "C[2]" "6# &%\"CG6#\"\"#" }{TEXT -1 70 " in the expression for the derivative of \+ the general solution, gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=-2*exp(-2/3*x)+8/9" "6#/*&%#dyG\"\"\"%#dxG!\"\",& *&\"\"#F&-%$expG6#,$*(F+F&\"\"$F(%\"xGF&F(F&F(*&\"\")F&\"\"*F(F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x*exp(-2/3*x)" "6#*&%\"xG\"\"\"-%$expG6 #,$*(\"\"#F%\"\"$!\"\"F$F%F-F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx=exp(-2*x/3)*(8*x/9-2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$*( \"\"#F&%\"xGF&\"\"$F(F(F&,&*(\"\")F&F0F&\"\"*F(F&F/F(F&" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Any turning point on the graph of the solution curve occurs where " } {XPPEDIT 18 0 "dy/dx=0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "This only happens when " } {XPPEDIT 18 0 "8*x/9-2=0" "6#/,&*(\"\")\"\"\"%\"xGF'\"\"*!\"\"F'\"\"#F *\"\"!" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "x=9/4" "6#/%\"xG *&\"\"*\"\"\"\"\"%!\"\"" }{XPPEDIT 18 0 " ``= 2.25" "6#/%!G-%&FloatG6$ \"$D#!\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The corresp onding " }{TEXT 304 1 "y" }{TEXT -1 15 " coordinate is " }{XPPEDIT 18 0 "y=-2*exp(-3/2)" "6#/%\"yG,$*&\"\"#\"\"\"-%$expG6#,$*&\"\"$F(F'! \"\"F/F(F/" }{TEXT -1 1 " " }{TEXT 323 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" }{TEXT -1 13 ".4462603203. " }} {PARA 0 "" 0 "" {TEXT -1 66 "The sign of the derivative is determined \+ by the sign of the factor" }{XPPEDIT 18 0 " ``(8*x/9-2)" "6#-%!G6#,&*( \"\")\"\"\"%\"xGF)\"\"*!\"\"F)\"\"#F," }{TEXT -1 23 " which is negativ e for " }{XPPEDIT 18 0 "x<9/4" "6#2%\"xG*&\"\"*\"\"\"\"\"%!\"\"" } {TEXT -1 18 " and positive for " }{XPPEDIT 18 0 "x>9/4" "6#2*&\"\"*\" \"\"\"\"%!\"\"%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "T he single turning point" }{XPPEDIT 18 0 " ``(9/4,2*exp(-3/2))" "6#-%!G 6$*&\"\"*\"\"\"\"\"%!\"\"*&\"\"#F(-%$expG6#,$*&\"\"$F(F,F*F*F(" } {TEXT -1 65 " on the graph of the solution must therefore be a minimum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "-2*exp(-3/2);\nevalf(evalf(%,13));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$expG6##!\"$F%F&!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+.KgiW!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "de := 9*diff(y(x),x$2)+12*diff(y(x),x)+4*y(x)=0;\nic := y(0)=1,D( y)(0)=-2;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%),x):\nD(g)(x); \n[solve(D(g)(x),x)];\nop(map(_x->[_x,g(_x)],%));\nevalf(evalf(%,13)); \nplot(g(x),x=0..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\" \"*\"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F0\"\"#F)F)*&\"#7F)-F+6$F-F0F) F)*&\"\"%F)F-F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-% \"yG6#\"\"!\"\"\"/--%\"DG6#F(F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"yG6#%\"xG,&-%$expG6#,$*(\"\"#\"\"\"\"\"$!\"\"F'F/F1F/*&#\"\"%F0F /*&F'F/F)F/F/F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"-%$e xpG6#,$*(F%F&\"\"$!\"\"%\"xGF&F-F&F-*&#\"\")\"\"*F&*&F.F&F'F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7##\"\"*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"\"*\"\"%,$*&\"\"#\"\"\"-%$expG6##!\"$F)F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"++++]A!\"*$!+.KgiW!#5" }}{PARA 13 "" 1 "" {GLPLOT2D 636 249 249 {PLOTDATA 2 "6%-%'CURVESG6$7Z7$$\"\"! F)$\"\"\"F)7$$\"3]mm;z>]9Q!#>$\"3_\\82Tk2`#*!#=7$$\"3+LLLeR+HwF/$\"3Y] KFhwNP&)F27$$\"3&****\\Pf]V9\"F2$\"3+\")*Hg^Z<&yF27$$\"3gmmm\"z+e_\"F2 $\"3'>owPB&=&>(F27$$\"3;+](oM'f*=#F2$\"3?+\\d*Gm)=hF27$$\"3sLL3->R`GF2 $\"3X#4'Hp#[A7&F27$$\"3=+]7G%**)*f$F2$\"3oV'yvy(f!4%F27$$\"3mmm;apSYVF 2$\"3AfP^_a/ZJF27$$\"3Gnmm;G'y4&F2$\"3m[//4\"=+G#F27$$\"3Onm;z'=$\\eF2 $\"3cR<%p'y?!\\\"F27$$\"3i+]7.I?(f'F2$\"3)GUe()*4!Rv(F/7$$\"3!RL$3Ft3X tF2$\"33_SWCdzl7F/7$$\"3tmmTNj&=t)F2$!3/zv[%f\"ow\"*F/7$$\"33+](=`xn, \"!#<$!3puEmz#Hf!=F27$$\"3#omT&y/Gl6F\\p$!3?Du)HHEia#F27$$\"3++]PurI88 F\\p$!3?'p[%*=n#HJF27$$\"3aLL$e#3dl9F\\p$!3/=wEA8T\"f$F27$$\"3ymm\"Ht% o*f\"F\\p$!31>xcvCy**QF27$$\"3K++]F_m]F\\p$!3!y#GpE1WAVF27$$\"3;++]s2O[?F\\p$!3R>b#[U![=WF27$$\"3um;aG\"H 5=#F\\p$!3#Q#\\iA'QxX%F27$$\"3^LL$ej%yQBF\\p$!3sMkXDx3bWF27$$\"3mLLLVU UsCF\\p$!3/]0XEC8=WF27$$\"35+](o()yyi#F\\p$!35zcP8inUVF27$$\"3GLLLoD[l FF\\p$!3+zTzhdM_UF27$$\"3P+](oibk\"HF\\p$!3z/b81k@LTF27$$\"3a+]i!o<-1$ F\\p$!3VcsR)[!o/SF27$$\"3qLL3-$=-@$F\\p$!3K=UZp'f)eQF27$$\"3kL$3xplzM$ F\\p$!3r-]FA'ytr$F27$$\"3gmm\"H([a'\\$F\\p$!3/FifU2OfNF27$$\"3wm;ayo(3 l$F\\p$!35g-fw_!=R$F27$$\"3?+]7VLA&y$F\\p$!3L%4qAn_[C$F27$$\"3'pm;a?@. $RF\\p$!3Q28ww@\\'3$F27$$\"3)******\\\\@-3%F\\p$!3]3)=Rv,Y#HF27$$\"3Q+ +v$opoA%F\\p$!3YPL3v/+pFF27$$\"3c+](oMf(oVF\\p$!3DPvys9)=i#F27$$\"3#)* **\\ii.j_%F\\p$!3m[?Z*zQKY#F27$$\"3%GLL$oT'ym%F\\p$!3esHU4'GaK#F27$$\" 3'3++DE5!>[F\\p$!3'e*R9A0k$=#F27$$\"3Mm;a)3rf&\\F\\p$!3![%3(yYw,1#F27$ $\"3*4++vW0d5&F\\p$!3F27$$\"3;L$3-\"QfY_F\\p$!3ycw(HW(e9=F27 $$\"3C+]PWF'QR&F\\p$!3qs:g=7x)p\"F27$$\"3[LL$e/Xy`&F\\p$!3[KkrT-9\"f\" F27$$\"3m**\\(=<\"e)o&F\\p$!3m;.EW:K%[\"F27$$\"3%ymmm(zvLeF\\p$!3!\\y; b'Q+(Q\"F27$$\"3-nm\"zAAA)fF\\p$!359wGsq)HH\"F27$$\"3LM$3-7d%HhF\\p$!3 J?T0JZ307F27$$\"3#4++]p]ZE'F\\p$!3-j?3EH%)G6F27$$\"3$QL$e*R7)>kF\\p$!3 Ojuh::gY5F27$$\"3'pmmmV,&elF\\p$!3;%46D!>7v(*F/7$$\"3<+](o(GP1nF\\p$!3 /%pM]BZH3*F/7$$\"3g+]78Z!z%oF\\p$!3!QR#)GB5:Y)F/7$$\"\"(F)$!3U6gCE@IOy F/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fg]l-%%VIEW G6$;F(Fh\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 38 "Consider the d ifferential equation 9 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+23/2;" "6#,&*( %\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&\"#BF(F&F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\" F'*&\"\"%F'%\"yGF'F'\"\"!" }{TEXT -1 51 ", which is obtained by changi ng the coefficient of " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG! \"\"" }{TEXT -1 67 " in the differential equation of the previous exam ple from 12 to 11" }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 78 ". The graph of the particular solution subject to the sam e initial conditions " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\" \"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`y '`(0) = -2;" "6#/-%$y~'G6#\" \"!,$\"\"#!\"\"" }{TEXT -1 47 " as before looks similar to the previou s graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "de := 9*diff(y(x),x$2)+23/2*diff(y(x),x)+4*y(x)=0;\n ic := y(0)=1,D(y)(0)=-2;\ndsolve(\{de,ic\},y(x));\ng := unapply(rhs(%) ,x);\nplot(g(x),x=0..16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,( *&\"\"*\"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F0\"\"#F)F)*&#\"#BF4F)-F+6 $F-F0F)F)*&\"\"%F)F-F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ic G6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"#\\\"#Z\"\"\"*(F,#F-\"\"#-%$expG6# ,$*(\"#BF-\"#O!\"\"F'F-F8F--%$sinG6#,$*(F7F8F,F/F'F-F-F-F-F8*&F1F--%$c osGF;F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,&*&#\"#\\\"#Z\"\"\"*(F0#F1\"\"#-%$expG6#,$*&#\"#B\"#O F19$F1!\"\"F1-%$sinG6#,$*&#F1F*&F5F1-%$cosGFAF1 F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 411 255 255 {PLOTDATA 2 "6%-%'C URVESG6$7]o7$$\"\"!F)$\"\"\"F)7$$\"39LLLL3VfV!#>$\"35*RmcfEz9*!#=7$$\" 3Hmmmm;')=()F/$\"3WdY*[>/XL)F27$$\"3%*******\\#HyI\"F2$\"3c9'\\4)oIevF 27$$\"3ELLLLBxV$F27$$\"3/+++];z/]F2$\"3!G99+m2+H#F27$$\"3!omm;9:Mw&F2$\"3-ZLRafo g9F27$$\"3cLLLL'Q?_'F2$\"3qPv'\\h7U/(F/7$$\"36+++]:MG#)F2$!3cs3st9]XvF /7$$\"3ommmmWkM**F2$!3([O&[V0&*=>F27$$\"3kmmm'yD_;\"!#<$!3/6g!3Bu#QGF2 7$$\"3jmmmEr)pL\"Fgo$!35x(QVpfIa$F27$$\"3-+++:@$z]\"Fgo$!3%R%z$eh#=lSF 27$$\"3;LLL.r()y;Fgo$!37pmYgqZPWF27$$\"3v*******)\\OP=Fgo$!3!RFgo$!3ohir([(*R\"[F27$$\"3u*******Q0z2#Fgo$!3%oAC'Q[ !z&[F27$$\"3%HLLL!z&*f@Fgo$!3]^RPr^k$)[F27$$\"37mmm;/,UAFgo$!3aYv3^O&H *[F27$$\"3v******HH1CBFgo$!3)QV;!*RRu)[F27$$\"3'GLLLe!y$\\#Fgo$!3#3J%) 4!eGN[F27$$\"3SmmmO#)\\jEFgo$!3Ejz3#y5xt%F27$$\"3h******p\\%=+$Fgo$!3] a8t*)QwXWF27$$\"3aLLLth()\\LFgo$!3s0efB*[?1%F27$$\"3smmmYAUcOFgo$!3aBV =:5W\"p$F27$$\"3p******>0_,SFgo$!3OrL]TvmmKF27$$\"3!********zN![VFgo$! 3GAao%Fgo$!3]BK*=q:wZ#F27$$\"3gmmmO%4_)\\Fgo $!3]g,)>rWO;#F27$$\"37LLL`MzX`Fgo$!3c1QnNZcE=F27$$\"3#GLLLTb7l&Fgo$!3I FrLS;bs:F27$$\"3g*******G!e1gFgo$!3())=Q)=[c78F27$$\"3eKLL8I5@jFgo$!3> n5bJK_76F27$$\"3M+++!H%=mmFgo$!3=$**R&\\j_H#*F/7$$\"35+++qKy%*pFgo$!3G 7&f:i%e(o(F/7$$\"3`LLLL=kPtFgo$!3#\\f](Hj-AjF/7$$\"3ELLLBI\\_wFgo$!3#z zE'eI&3E&F/7$$\"3_mmmmD5#*zFgo$!3w-lx`R\\'H%F/7$$\"3NlmmO9'[M)Fgo$!3)4 F#ez(fbY$F/7$$\"3=******p!R>l)Fgo$!3%*)fo#[#4N'GF/7$$\"3#emmmK\"f$)*)F go$!3P\"[0w9>7K#F/7$$\"3W******f0AE$*Fgo$!3)=F6Vd!yg=F/7$$\"3M)*****>k Th'*Fgo$!3')39.6Rl#\\\"F/7$$\"3u)*****\\ct&)**Fgo$!3A=zhd/?,7F/7$$\"3' )*****fo$eM5!#;$!3WE*Qt!4*QR*!#?7$$\"3ALLL\"QSp1\"Fbz$!39:YoKLo*\\(Fez 7$$\"3(*******f!)[,6Fbz$!3;,x,eoOpeFez7$$\"3gmmm\"R$zK6Fbz$!3w7_kTk!*z YFez7$$\"3)******zQ=q;\"Fbz$!3(G&e,9CANOFez7$$\"3;LLLU9A*>\"Fbz$!3G**) Q$osf^GFez7$$\"3\"******H\"H)GB\"Fbz$!3I&Q8:qA)*>#Fez7$$\"3GLLL`Jzl7Fb z$!37@\"*))f%*Q'p\"Fez7$$\"3$******\\7Z-I\"Fbz$!3a9kL7x+$G\"Fez7$$\"3r mmm%RIML\"Fbz$!37at,wVUF(*!#@7$$\"3immm!3ltO\"Fbz$!35#zv\"*=CEE(Fc]l7$ $\"3JLLLq(=5S\"Fbz$!3MB`pFyVy`Fc]l7$$\"3'******f,V>V\"Fbz$!3\"3=$G#f`o .%Fc]l7$$\"3KLLL\"p&Qn9Fbz$!3!Q(=7<+NgGFc]l7$$\"3gmmmUg3*\\\"Fbz$!3%fD &**z7-l?Fc]l7$$\"3/+++H_)G`\"Fbz$!3UTny9\"\\XU\"Fc]l7$$\"3/+++j`Bl:Fbz $!3KF&e*4.*on*!#A7$$\"#;F)$!3utIYsuKugFg_l-%'COLOURG6&%$RGBG$\"#5!\"\" F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fh`l-%%VIEWG6$;F(Fi_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "However, it is c lear from the formula for the solution that it is oscillatory. Investi gating some other graphs shows some of the oscillations and enables us to find some more " }{TEXT 324 1 "x" }{TEXT -1 13 " intercepts. " }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 326 1 "x" }{TEXT -1 25 " inte rcepts are given by " }{XPPEDIT 18 0 "x=36/sqrt(47)" "6#/%\"xG*&\"#O\" \"\"-%%sqrtG6#\"#Z!\"\"" }{XPPEDIT 18 0 "``(arctan(sqrt(47)/49)+Pi*k); " "6#-%!G6#,&-%'arctanG6#*&-%%sqrtG6#\"#Z\"\"\"\"#\\!\"\"F/*&%#PiGF/% \"kGF/F/" }{TEXT -1 8 ", where " }{TEXT 325 1 "k" }{TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "_EnvAllSolutions := true:\nsolve(g(x)=0,x);\n_EnvAl lSolutions := false:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"#O\"#Z \"\"\"*&,&-%'arctanG6#,$*&\"#\\!\"\"F'#F(\"\"#F(F(*&%#PiGF(%%_Z6|irGF( F(F(F'F2F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "fsolve(g(x)=0,x=1);\n36/sqrt(47)*(arctan(sqrt(47 )/49));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+j[b*H(!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"#O\"#Z\"\"\"*&-%'arctanG6#,$*& \"#\\!\"\"F'#F(\"\"#F(F(F'F1F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+l[b*H(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot(g(x),x=16..32);\nfsolve(g(x)=0,x=17);\n36/s qrt(47)*(arctan(sqrt(47)/49)+Pi);\nevalf(%);" }}{PARA 13 "" 1 "" {GLPLOT2D 431 253 253 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"#;\"\"!$!3utI YsuKug!#A7$$\"3cmmmh)=(3;!#;$!3\\Kd4m(=OM&F-7$$\"36LLLBxV<;F1$!3=fTskf 'Gn%F-7$$\"3n*****\\echi\"F1$!3'G+'z#3#*y0%F-7$$\"3emmmYa([j\"F1$!33/L e6Ay%\\$F-7$$\"3>LLe\"ohCk\"F1$!3FZ#G0#z:WIF-7$$\"3;++];z/];F1$!3V:o,. `uFEF-7$$\"3ymmT^Tjd;F1$!3kRv5/')RVAF-7$$\"3QLLL'Q?_m\"F1$!3sOm__@4*)= F-7$$\"3E++]:MG#o\"F1$!3%>:B!p=B#>\"F-7$$\"3wmmmWkM*p\"F1$!3y\"fI\"\\! y.='!#B7$$\"3tmmmyD_;F1$\"3'Hm**Qgq#y7F-7$$\"3`LLLF1$\"3n$zJ QL3S@\"F-7$$\"3]mmmCAkl>F1$\"3VQv/t7[K6F-7$$\"3(******>0_,+#F1$\"3Fb&H YLYg-\"F-7$$\"3#)******zN![.#F1$\"3&eo?F1$\"3? s@4$zJy0)F[o7$$\"3[mmmV4_)4#F1$\"3uNi7qtLBrF[o7$$\"3BLLLX$zX8#F1$\"3in :2*pK24'F[o7$$\"35LLLTb7l@F1$\"37,jV@@K%H&F[o7$$\"3()******G!e1?#F1$\" 3-e!fMvRQY%F[o7$$\"3xh$e%3>$F[o7$$\"3#******pKy%*H#F1$\"3)=!H35USxEF[o7$$\"3FLLL$=kPL# F1$\"3v&*GsAE$y@#F[o7$$\"3]LLL-$\\_O#F1$\"3q!3Hg#\\Jd=F[o7$$\"3mmmmc-@ *R#F1$\"3wLm[!p!)o_\"F[o7$$\"3ammmVh[MCF1$\"3Id$*z1(y(R7F[o7$$\"3u**** *p!R>lCF1$\"3\\Cu0m=?I5F[o7$$\"3emmmK\"f$)\\#F1$\"308P.l>X,%)!#C7$$\"3 &******f0AE`#F1$\"33^N(H&RqwnF]x7$$\"3%)*****>kThc#F1$\"3i<(e_G(4paF]x 7$$\"3))*****\\ct&)f#F1$\"3!)HE2b[PFWF]x7$$\"3')*****fo$eMEF1$\"3]H4& \\b.d[$F]x7$$\"3ALLL\"QSpm#F1$\"3&p)oO7`9+GF]x7$$\"3z******f!)[,FF1$\" 3?#4R(Q6`1AF]x7$$\"3gmmm\"R$zKFF1$\"3-FOv.;%3x\"F]x7$$\"3)******zQ=qw# F1$\"3gx5gFF*eQ\"F]x7$$\"3)HLLBW@#*z#F1$\"3=*3blvY`4\"F]x7$$\"3\"***** *H\"H)G$GF1$\"3oFO,dLl@&)!#D7$$\"3GLLL`JzlGF1$\"34s;%**>+4j'F`[l7$$\"3 $******\\7Z-!HF1$\"3+4I!)=A`n]F`[l7$$\"3`mmm%RIM$HF1$\"3A$oK]'[W&)QF`[ l7$$\"3Ymmm!3lt'HF1$\"3'z@R9GG!RHF`[l7$$\"3JLLLq(=5+$F1$\"3n*))H$3JS4A F`[l7$$\"3'******f,V>.$F1$\"3#oPY(G4H&o\"F`[l7$$\"39LLL\"p&QnIF1$\"3&* RZ&o_+6A\"F`[l7$$\"3WmmmUg3*4$F1$\"3DG#eABQu.*!#E7$$\"3/+++H_)G8$F1$\" 3uD4n/(H*\\kFi]l7$$\"3/+++j`BlJF1$\"3Y>Ii9hLxXFi]l7$$\"#KF*$\"33obRT)= ^2$Fi]l-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F`_l-%+AXESLABELSG6$Q\"x6\"Q!F e_l-%%VIEWG6$;F(Fe^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+P(*oA " 0 "" {MPLTEXT 1 0 91 "plot(g(x),x=32..48); 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" }}}{PARA 0 "" 0 "" {TEXT -1 129 "For each of questions 1 to 3 in whi ch an homogeneous linear 2nd order differential equations with constan t coefficients is given:" }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Set up th e auxiliary equation \"by hand\" and solve it using " }{TEXT 0 5 "solv e" }{TEXT -1 15 ", if necessary." }}{PARA 0 "" 0 "" {TEXT -1 65 "(b) F ind the general solution of the differential equation using " }{TEXT 0 6 "dsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "(c) Check \+ that the solution given by " }{TEXT 0 6 "dsolve" }{TEXT -1 83 " is con sistent with the 3 cases described above, and state which case is invo lved: " }}{PARA 0 "" 0 "" {TEXT -1 37 " (i) the auxiliary equatio n has " }{TEXT 261 23 "distinct real solutions" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 37 " (ii) the auxiliary equation has " } {TEXT 261 20 "exactly one solution" }{TEXT -1 22 " (or equal solutions )," }}{PARA 0 "" 0 "" {TEXT -1 36 "or (iii) the auxiliary equation has " }{TEXT 261 17 "complex solutions" }{TEXT -1 71 " (strictly speaking , complex solutions with a non-zero imaginary part)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 7 "Example" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 " Consider" }}{PARA 256 "" 0 "" {TEXT -1 4 " 2 " }{XPPEDIT 18 0 "d^2*y/(d*x^2) + 5" "6#,&*(%\"dG\"\"#%\"yG \"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-3*y = 0" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F)\"\"! " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) The auxiliary equation is " } {XPPEDIT 18 0 "2*m^2+5*m-3=0" "6#/,(*&\"\"#\"\"\"*$%\"mGF&F'F'*&\"\"&F 'F)F'F'\"\"$!\"\"\"\"!" }{TEXT -1 33 ". The left side factors to give \+ " }{XPPEDIT 18 0 "(2*m-1)*(m+3)=0" "6#/*&,&*&\"\"#\"\"\"%\"mGF(F(F(! \"\"F(,&F)F(\"\"$F(F(\"\"!" }{TEXT -1 23 ", so the solutions are " } {XPPEDIT 18 0 "m=1/2" "6#/%\"mG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m=-3" "6#/%\"mG,$\"\"$!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "2*m^2+5*m-3 = 0;\nfactor(%);\nsolve(%,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"mG\"\"#\"\"\"F(*&\"\"&F)F'F)F)\"\"$!\"\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"mG\"\"\"\"\"$F'F',&F&\"\"# F'!\"\"F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"\"#!\"$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "de := 2*diff(y(x),x$2)+5*dif f(y(x),x)-3*y(x)=0;\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#F1*&\"\"&\"\"\"-F(6$ F*F-F4F4*&\"\"$F4F*F4!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,&*&%$_C1G\"\"\"-%$expG6#,$F'!\"$F+F+*&%$_C2GF+-F-6#,$F'#F+ \"\"#F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "(c) The case of " }{TEXT 261 23 "distinct real solutions" } {TEXT -1 56 " for the auxiliary equation is involved in this example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 258 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+5;" "6#,&*(%\"d G\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"&F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+6*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"'F '%\"yGF'F'\"\"!" }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "_ ___________________________________________ " }}{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 45 "_______________________________ _____________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 258 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-4;" "6#,&*(%\"dG\"\"#%\"yG\"\" \"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx+4*y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F'\"\"!" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________ _____________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 258 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+5*y = 0;" "6#/,&*&%# dyG\"\"\"%#dxG!\"\"F'*&\"\"&F'%\"yGF'F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "____________________________________________ " }}{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 45 "__ __________________________________________ " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 258 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"! " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "_____________________ _______________________ " }}{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 45 "____________________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 258 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+9*y = 0;" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\" xGF'F)!\"\"F)*&\"\"*F)F(F)F)\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "____________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________ ________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }} {PARA 258 "" 0 "" {TEXT -1 7 " 3 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+d y/dx-2*y = 0" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&% #dyGF)%#dxGF-F)*&F'F)F(F)F-\"\"!" }{TEXT -1 4 " " }}{PARA 0 "" 0 " " {TEXT -1 45 "____________________________________________ " }}{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 45 "____________ ________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }} {PARA 258 "" 0 "" {TEXT -1 8 " 2 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+ 6;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"'F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+5*y = 0;" "6#/,&*&%#dyG\"\"\"%#dx G!\"\"F'*&\"\"&F'%\"yGF'F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "____________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________ ________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }} {PARA 258 "" 0 "" {TEXT -1 64 "(a) Find the particular solution of the differential equation " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-dy/dx-2*y = 0 " "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF -F-*&F'F)F(F)F-\"\"!" }{TEXT -1 40 ", which satisfies the intial condi tions " }{XPPEDIT 18 0 "y(0) = 4;" "6#/-%\"yG6#\"\"!\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = -1;" "6#/*&%\"yG\"\"\"-%\"'G6#\" \"!F&,$F&!\"\"" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 36 "(b) P lot the graph of the solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 24 ": The initial conditi on " }{XPPEDIT 18 0 "y*`'`(0) = -1;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&, $F&!\"\"" }{TEXT -1 20 " should be given to " }{TEXT 0 6 "dsolve" } {TEXT -1 18 " as the equation: " }{TEXT 262 10 "D(y)(0)=-1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "_________________________________ ___________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 45 "____________________________________________ " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q9 " }}{PARA 258 "" 0 "" {TEXT -1 65 "(a) Find the parti cular solution of the differential equation 4 " }{XPPEDIT 18 0 "d^2*y /(d*x^2)+12;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"#7 F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+9*y = 0;" "6#/,&*&%#dyG\"\" \"%#dxG!\"\"F'*&\"\"*F'%\"yGF'F'\"\"!" }{TEXT -1 40 ", which satisfies the intial conditions " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y*`'`(0) = 1" "6#/*&%\"yG\"\"\"-% \"'G6#\"\"!F&F&" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 36 "(b) \+ Plot the graph of the solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 24 ": The initial conditi on " }{XPPEDIT 18 0 "y*`'`(0) = 1" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&F& " }{TEXT -1 20 " should be given to " }{TEXT 0 6 "dsolve" }{TEXT -1 18 " as the equation: " }{TEXT 262 9 "D(y)(0)=1" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 45 "_________________________________________ ___ " }}{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 45 "____________________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q10 " }}{PARA 258 "" 0 "" {TEXT -1 62 "(a) Find the particular soluti on of the differential equation " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+dy/dx+ 33/4;" "6#,(*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&%#dyGF(%# dxGF,F(*&\"#LF(\"\"%F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "y = 0;" "6#/ %\"yG\"\"!" }{TEXT -1 40 ", which satisfies the intial conditions " } {XPPEDIT 18 0 "y(0)=2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y*`'`(0) = 0" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&F*" } {TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 36 "(b) Plot the graph of \+ the solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 24 ": The initial condition " }{XPPEDIT 18 0 "y*`'`(0) = 0;" "6#/*&%\"yG\"\"\"-%\"'G6#\"\"!F&F*" }{TEXT -1 20 " should be given to " }{TEXT 0 6 "dsolve" }{TEXT -1 18 " as the equat ion: " }{TEXT 262 9 "D(y)(0)=0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "____________________________________________ " }}{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 45 "____________ ________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 }