{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 261 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 261 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 274 "Times" 1 12 96 52 84 1 0 1 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 "Hea ding 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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Time s" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 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 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -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 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 41 "Laplace transforms of piecewise f unctions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C. , Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The unit step function " }{XPPEDIT 18 0 " u[a](t);" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 28 " and its Laplace tra nsform: " }{XPPEDIT 18 0 "L*[u[a](t)] = exp(-a*s)/s;" "6#/*&%\"LG\"\" \"7#-&%\"uG6#%\"aG6#%\"tGF&*&-%$expG6#,$*&F,F&%\"sGF&!\"\"F&F5F6" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "For each positive real number " }{TEXT 273 1 "a" }{TEXT -1 13 " we define a " }{TEXT 259 18 "unit step function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u[a](t);" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 3 " b y" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "u[a](t) = PIEC EWISE([0, t < a],[1, a <= t]);" "6#/-&%\"uG6#%\"aG6#%\"tG-%*PIECEWISEG 6$7$\"\"!2F*F(7$\"\"\"1F(F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[a](t);" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 2 " \"" }{TEXT 259 11 "switches on" }{TEXT -1 22 "\" to the value 1 when " }{XPPEDIT 18 0 "t = a" "6#/%\"tG%\"aG" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 396 199 199 {PLOTDATA 2 "6 /-%'CURVESG6%7$7$$\"\"!F)F(7$$\"\"#F)F(-%'COLOURG6&%$RGBG$\"*++++\"!\" )F(F(-%*THICKNESSG6#F,-F$6%7$7$F+$\"\"\"F)7$$\"\"%F)F;F-F4-F$6&7#F*F-- %'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-F$6&7#F:F-FCFG-F$6&FMF--FD6#%( DIAMONDGFG-F$6&FMF--FD6#%&CROSSGFG-%%TEXTG6&7$F+$!#8!\"#Q\"a6\"-%&COLO RG6&F0$F$!\"\"FhoQ\"tFjnF [oF_o-FY6&7$$FhnFho$\"#:FhoQ'u~~(t)FjnF[oF_o-FY6&7$$!#BFhn$\"$X\"FhnFi nF[o-F`o6$Fbo\"\"*-%*AXESTICKSG6$F)\"\"$-%+AXESLABELSG6%Q!FjnFbq-F`o6# %(DEFAULTG-%%VIEWG6$;FdpF>;Ffn$\"#>Fho" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 9 "Heavi side" }{TEXT -1 26 " function H is defined by " }{XPPEDIT 18 0 "H(t)=u [0](t)" "6#/-%\"HG6#%\"tG-&%\"uG6#\"\"!6#F'" }{XPPEDIT 18 0 "``= PIECE WISE([0, t < 0],[1, 0 <= t])" "6#/%!G-%*PIECEWISEG6$7$\"\"!2%\"tGF)7$ \"\"\"1F)F+" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 397 199 199 {PLOTDATA 2 "6--%'CURVESG6%7$7$$!\"#\"\"!$F*F*7$ F+F+-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"\"#-F$6%7$7$F +$\"\"\"F*7$$F7F*FFgn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "u[ a](t)=H(t-a)" "6#/-&%\"uG6#%\"aG6#%\"tG-%\"HG6#,&F*\"\"\"F(!\"\"" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L* [u[a](t)];" "6#*&%\"LG\"\"\"7#-&%\"uG6#%\"aG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(u[a](t)*exp(-s*t),t = 0 .. infinity);" "6#-%$In tG6$*&-&%\"uG6#%\"aG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF.F-F.!\"\"F./F-;\" \"!%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " } {XPPEDIT 18 0 "Int(u[a](t)*exp(-s*t),t = 0 .. a);" "6#-%$IntG6$*&-&%\" uG6#%\"aG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF.F-F.!\"\"F./F-;\"\"!F+" } {TEXT -1 4 " + " }{XPPEDIT 18 0 "Int(u[a](t)*exp(-s*t),t = a .. infin ity);" "6#-%$IntG6$*&-&%\"uG6#%\"aG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF.F- F.!\"\"F./F-;F+%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "u[a](t);" "6#-&%\"uG6#%\"aG6#%\"tG" } {TEXT -1 46 " = 0 throughout the (open) interval from 0 to " }{TEXT 270 1 "a" }{TEXT -1 38 ", the first integral is 0, and since " } {XPPEDIT 18 0 "u[a](t);" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 34 " = 1 \+ throughout the interval from " }{TEXT 269 1 "a" }{TEXT -1 21 " to infi nity, we have" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "L*[u[a](t)];" "6#*&% \"LG\"\"\"7#-&%\"uG6#%\"aG6#%\"tGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(exp(-s*t),t = a .. infinity);" "6#-%$IntG6$-%$expG6#,$*&%\"sG\"\" \"%\"tGF,!\"\"/F-;%\"aG%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Limit(``,R = infinity);" "6#/%!G-%&LimitG6$F$/%\"RG%)infinityG" } {XPPEDIT 18 0 "exp(-s*t)/(-s);" "6#*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF*! \"\"F*,$F)F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[a, \+ ``]);" "6#-%*PIECEWISEG6$7$%\"RG%!G7$%\"aGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " } {XPPEDIT 18 0 "Limit(``(exp(-s*R)/(-s)+exp(-s*a)/s),R = infinity);" "6 #-%&LimitG6$-%!G6#,&*&-%$expG6#,$*&%\"sG\"\"\"%\"RGF1!\"\"F1,$F0F3F3F1 *&-F,6#,$*&F0F1%\"aGF1F3F1F0F3F1/F2%)infinityG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "exp(-s*a)/s;" "6#* &-%$expG6#,$*&%\"sG\"\"\"%\"aGF*!\"\"F*F)F," }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following Maple procedure " }{TEXT 0 1 "u" } {TEXT -1 31 " implements the step function " }{XPPEDIT 18 0 "u[a](t); " "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "u := proc(t)\n \+ local a;\n if type(procname,specindex(algebraic,u)) and type(t,algeb raic) then\n a := op(1,procname); \n piecewise(t=a,1 );\n else 'procname'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "'u[a](t)'=u[ a](t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"uG6#%\"aG6#%\"tG-%*PIE CEWISEG6$7$\"\"!2F*F(7$\"\"\"1F(F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot(u[2](t),t=0..4,0..1.3, discont=true,thickness=2,color=red,ytickmarks=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 500 175 175 {PLOTDATA 2 "6'-%'CURVESG6&7S7$$\"\"!F)F(7$$\"3S rWhC3VfV!#>F(7$$\"3Sp:Ov#[D:)F-F(7$$\"39s'\\ebI=C\"!#=F(7$$\"3gl3*\\!R Br;F4F(7$$\"3et%p\\P'f)4#F4F(7$$\"3'3qV34;[\\#F4F(7$$\"3LU)*oc'y]!HF4F (7$$\"3z(eu\"*ys$HLF4F(7$$\"3/(Q&*\\?1Bv$F4F(7$$\"3aw>H3_M(=%F4F(7$$\" 3qxA>*zF0d%F4F(7$$\"3\\)>'**R1!>+&F4F(7$$\"3g5*H\"RZ/NaF4F(7$$\"3!=3&H Q$fC&eF4F(7$$\"3J4.P$y6:B'F4F(7$$\"3U#=-L!=C#o'F4F(7$$\"3OF&QDDpS1(F4F (7$$\"3>\\N[Z`A3vF4F(7$$\"3t2R'3vy8!zF4F(7$$\"3?RX$eMIFL)F4F(7$$\"3fTI ,q!zMu)F4F(7$$\"3yhDKtA0s\"*F4F(7$$\"3UMa.gihl&*F4F(7$$\"3qvIN)=G,***F 4F(7$$\"3(z6(\\xw5V5!#Q**H!*3`i\"FapF(7$$ \"3Id(****)zym;FapF(7$$\"3h?\\TZj?4TTfMF^FapF(7$$\"3_6p$)*>Wl&>FapF(7$$\"3))*****f*******>FapF(7S7$$\"3A++ +3+++?Fap$\"\"\"F)7$$\"3[c*ehJ%fV?FapF[u7$$\"37l0f!\\D:3#FapF[u7$$\"3B ,mLjI=C@FapF[u7$$\"3!*R[;)RBr;#FapF[u7$$\"3KGs2X'f)4AFapF[u7$$\"3-2ae; ;[\\AFapF[u7$$\"3))oz3ty]!H#FapF[u7$$\"3Z%e^iGPHL#FapF[u7$$\"3yx!\\xiI _P#FapF[u7$$\"3sG<*z_M(=CFapF[u7$$\"3UA^+(y_qX#FapF[u7$$\"3=S#**42!>+D FapF[u7$$\"38#)f#3[/Na#FapF[u7$$\"3[;!f1%fC&e#FapF[u7$$\"3`G24&=^Ji#Fa pF[u7$$\"3;qP*p=C#oEFapF[u7$$\"3JR5%=$pS1FFapF[u7$$\"3+5nCTD#3v#FapF[u 7$$\"3I:h]\")y8!z#FapF[u7$$\"3B3p\"4/tK$GFapF[u7$$\"3')3ED8zMuGFapF[u7 $$\"3*f%y\\L_?*[H7$FapF[u7$$ \"3!)*)oLtvxlJFapF[u7$$\"3t\"Hpc0xw?$FapF[u7$$\"3iKrDfp@[KFapF[u7$$\"3 ]\"3F.hHKH$FapF[u7$$\"3tk>LzanLLFapF[u7$$\"3ofD\\x+'oP$FapF[u7$$\"3Rm$ >>u\"*fT$FapF[u7$$\"3A3\\;()HxeMFapF[u7$$\"3Gf0#\\!o-*\\$FapF[u7$$\"3M &e&3VO5TNFapF[u7$$\"3+,xLV9C#e$FapF[u7$$\"3ek([x!*3`i$FapF[u7$$\"3m\"= mY*zymOFapF[u7$$\"3R3l*>N1#4PFapF[u7$$\"3-t:\"RYt7v$FapF[u7$$\"37&GS3x G**y$FapF[u7$$\"3pQ(H[6KU$QFapF[u7$$\"3?.z$QbdQ(QFapF[u7$$\"3GubeOl5;R FapF[u7$$\"3mBQ#R?Wl&RFapF[u7$$\"\"%F)F[u-%'COLOURG6&%$RGBG$\"*++++\"! \")F(F(-%*THICKNESSG6#\"\"#-%'POINTSG6$7$$Fg^lF)F[uF]^l-%*AXESTICKSG6$ %(DEFAULTGFg^l-%+AXESLABELSG6$Q\"t6\"Q!Fe_l-%%VIEWG6$;F(F[^l;F($\"#8! \"\"" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The Maple funct ion " }{TEXT 0 9 "Heaviside" }{TEXT -1 43 ", is slightly different fr om the function " }{XPPEDIT 18 0 "H(t)" "6#-%\"HG6#%\"tG" }{TEXT -1 44 " defined above in that it has no value when " }{XPPEDIT 18 0 "t=0 " "6#/%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Heaviside(t) = PIECEW ISE([0, t < 0],[undefined, t = 0],[1, 0 < t]);" "6#/-%*HeavisideG6#%\" tG-%*PIECEWISEG6%7$\"\"!2F'F,7$%*undefinedG/F'F,7$\"\"\"2F,F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "For any number a with " }{XPPEDIT 18 0 "0 <= a;" "6#1\"\"!%\"aG" } {TEXT -1 14 " the function " }{XPPEDIT 18 0 "Heaviside(t-a);" "6#-%*He avisideG6#,&%\"tG\"\"\"%\"aG!\"\"" }{TEXT -1 30 " has the piecewise de scription" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Heaviside(t-a) = PIECEWISE([0, t < a],[undefi ned, t = a],[1, a < t]);" "6#/-%*HeavisideG6#,&%\"tG\"\"\"%\"aG!\"\"-% *PIECEWISEG6%7$\"\"!2F(F*7$%*undefinedG/F(F*7$F)2F*F(" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 21 "convert(..,Heaviside)" }{TEXT -1 56 " ignores the disitinction when used in conjunction with " }{TEXT 274 7 "u[a](t )" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 69 "'u[2](t)'=convert(u[2](t),Heaviside);\n``=co nvert(rhs(%),piecewise,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"uG 6#\"\"#6#%\"tG-%*HeavisideG6#,&F(!\"\"F*\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6%7$\"\"!2%\"tG\"\"#7$%*undefinedG/F+F ,7$\"\"\"2F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "laplace" }{TEXT -1 39 " can o nly handle expressions involving " }{TEXT 0 9 "Heaviside" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "inttrans[laplace](Heaviside(t-2),t,s);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*&\"\"#\"\"\"%\"sGF*!\"\"F*F+F," }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Other fo rms must be converted to expressions involving " }{TEXT 0 9 "Heaviside " }{TEXT -1 17 " before applying " }{TEXT 0 7 "laplace" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "'u[2](t)';\nconvert(%,Heaviside);\ninttrans[laplace]( %,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-&%\"uG6#\"\"#6#%\"tG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HeavisideG6#,&\"\"#!\"\"%\"tG\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*&\"\"#\"\"\"%\"sGF*! \"\"F*F+F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "The seco nd shift formula: " }{XPPEDIT 18 0 "L*[u[a](t)*f(t)] = exp(-a*s)*L*[f( t+a)];" "6#/*&%\"LG\"\"\"7#*&-&%\"uG6#%\"aG6#%\"tGF&-%\"fG6#F/F&F&*(-% $expG6#,$*&F-F&%\"sGF&!\"\"F&F%F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[u[a](t)*f(t)];" "6#*&%\"LG\"\"\"7#*& -&%\"uG6#%\"aG6#%\"tGF%-%\"fG6#F.F%F%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(u[a](t)*f(t)*exp(-s*t),t = 0 .. infinity);" "6#-%$IntG6$*(-& %\"uG6#%\"aG6#%\"tG\"\"\"-%\"fG6#F-F.-%$expG6#,$*&%\"sGF.F-F.!\"\"F./F -;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 6 " \+ = " }{XPPEDIT 18 0 "Int(u[a]*f(t)*exp(-s*t),t = 0 .. a);" "6#-%$IntG 6$*(&%\"uG6#%\"aG\"\"\"-%\"fG6#%\"tGF+-%$expG6#,$*&%\"sGF+F/F+!\"\"F+/ F/;\"\"!F*" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Int(u[a](t)*f(t)*exp(-s* t),t = a .. infinity);" "6#-%$IntG6$*(-&%\"uG6#%\"aG6#%\"tG\"\"\"-%\"f G6#F-F.-%$expG6#,$*&%\"sGF.F-F.!\"\"F./F-;F+%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "u[a](t) = 0; " "6#/-&%\"uG6#%\"aG6#%\"tG\"\"!" }{TEXT -1 73 " throughout the interv al from 0 to a, the first integral is 0, and since " }{XPPEDIT 18 0 "u [a](t) = 1;" "6#/-&%\"uG6#%\"aG6#%\"tG\"\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "a <= t;" "6#1%\"aG%\"tG" }{TEXT -1 24 ", the second int egral is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)* exp(-s*t),t = a .. infinity);" "6#-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$exp G6#,$*&%\"sGF+F*F+!\"\"F+/F*;%\"aG%)infinityG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 25 "We make the substitution " }{XPPEDIT 18 0 "tau = t-a;" "6#/%$tauG,&%\"tG\"\"\"%\"aG!\"\"" }{TEXT -1 26 " in th is integral so that " }{XPPEDIT 18 0 "d*tau = dt;" "6#/*&%\"dG\"\"\"%$ tauGF&%#dtG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t = tau+a;" "6#/%\"tG ,&%$tauG\"\"\"%\"aGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "Note that when " }{XPPEDIT 18 0 "t = a;" "6#/%\"tG%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tau = 0;" "6#/%$tauG\"\"!" }{TEXT -1 8 " and as \+ " }{XPPEDIT 18 0 "proc (t) options operator, arrow; infinity end;" "6# f*6#%\"tG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "proc (tau) options operator, arrow; infinity end;" " 6#f*6#%$tauG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 5 ", so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[u[a](t)* f(t)];" "6#*&%\"LG\"\"\"7#*&-&%\"uG6#%\"aG6#%\"tGF%-%\"fG6#F.F%F%" } {TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(f(t)*exp(-s*t),t = a .. infinity );" "6#-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$expG6#,$*&%\"sGF+F*F+!\"\"F+/F *;%\"aG%)infinityG" }{TEXT -1 3 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(tau+a)*exp(-s*tau-s*a),tau = 0 .. infini ty);" "6#/%!G-%$IntG6$*&-%\"fG6#,&%$tauG\"\"\"%\"aGF.F.-%$expG6#,&*&% \"sGF.F-F.!\"\"*&F5F.F/F.F6F./F-;\"\"!%)infinityG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-s*a)*Int(f( tau+a)*exp(-s*tau),tau = 0 .. infinity);" "6#/%!G*&-%$expG6#,$*&%\"sG \"\"\"%\"aGF,!\"\"F,-%$IntG6$*&-%\"fG6#,&%$tauGF,F-F,F,-F'6#,$*&F+F,F7 F,F.F,/F7;\"\"!%)infinityGF," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-s*a)*Int(f(t+a)*exp(-s*t),t = 0 .. infinity)" "6#/%!G*&-%$expG6#,$*&%\"sG\"\"\"%\"aGF,!\"\"F,-%$Int G6$*&-%\"fG6#,&%\"tGF,F-F,F,-F'6#,$*&F+F,F7F,F.F,/F7;\"\"!%)infinityGF ," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 71 "since it doesn't ma tter what variable is used in the definite integral." }}{PARA 0 "" 0 " " {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L*[u[a](t)*f(t)]=exp(-s*a)*L*[f(t+a)]" "6#/*&%\"LG\"\" \"7#*&-&%\"uG6#%\"aG6#%\"tGF&-%\"fG6#F/F&F&*(-%$expG6#,$*&%\"sGF&F-F&! \"\"F&F%F&7#-F16#,&F/F&F-F&F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 19 "___________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 7 "laplace" }{TEXT -1 65 " comes up with this formula \+ provided that we make the assumption " }{XPPEDIT 18 0 "0 < a;" "6#2\" \"!%\"aG" }{TEXT -1 17 " and reformulate " }{XPPEDIT 18 0 "u[a](t)*f(t )" "6#*&-&%\"uG6#%\"aG6#%\"tG\"\"\"-%\"fG6#F*F+" }{TEXT -1 37 " in ter ms of the Heaviside function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "f := 'f': \ninterface(showa ssumed=0):\nassume(a>0):\nHeaviside(t-a)*f(t);\n`Laplace transform`=in ttrans[laplace](%,t,s);\na := 'a':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&-%*HeavisideG6#,&%\"tG\"\"\"%#a|irG!\"\"F)-%\"fG6#F(F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*&-%$expG6#,$*&%\"sG\"\"\"% #a|irGF,!\"\"F,-%(laplaceG6%-%\"fG6#,&%\"tGF,F-F,F6F+F," }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "Examples of finding Laplace tran sforms using the second shift formula " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplace transform of the fun ction " }{XPPEDIT 18 0 "f(t)=u[2](t)*(t-2)^2" "6#/-%\"fG6#%\"tG*&-&%\" uG6#\"\"#6#F'\"\"\"*$,&F'F/F-!\"\"F-F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "Note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t)=PIECEWISE([0,t<2],[(t-2)^2,t>=2])" "6#/-%\"fG6#%\" tG-%*PIECEWISEG6$7$\"\"!2F'\"\"#7$*$,&F'\"\"\"F.!\"\"F.1F.F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "f := t -> (t-2)^2:\np1 := plot(f(t),t=2..4,thickness =2,color=red):\np2 := plot([[0,0],[2,0]],thickness=2,color=red):\nplot s[display]([p1,p2],tickmarks=[4,3],\n view=[0..4,0..4],labels=[ t,`f(t)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 205 185 185 {PLOTDATA 2 "6'- %'CURVESG6%7S7$$\"\"#\"\"!$F*F*7$$\"3ALLL3VfV?!#<$\"3?E<1>PY+>!#?7$$\" 3smm\"H[D:3#F/$\"3!yr&zkVSYmF27$$\"3XLL$e0$=C@F/$\"3<\"[5c8V@a\"!#>7$$ \"3QLL$3RBr;#F/$\"375JOwF-$z#F=7$$\"3%om;zjf)4AF/$\"3,^9lin5/WF=7$$\"3 WLLe4;[\\AF/$\"3X!3H?N2TA'F=7$$\"3-++Dmy]!H#F/$\"3t*y7`.#[R%)F=7$$\"3> LLezs$HL#F/$\"3u0NO@BZ36!#=7$$\"31++D@1BvBF/$\"3&*4mB\">!)zS\"FW7$$\"3 \"pmm;_M(=CF/$\"3YBTN'*fQ`+DF/$\"3*=/Z7,,>]#FW7$$\"3w*****\\Z/Na#F/$\"3e`-XV6(R&HFW7 $$\"35+++NfC&e#F/$\"3wV-MW!G^U$FW7$$\"36LLez6:BEF/$\"3)[*4;ER<$)QFW7$$ \"3_mmm\"=C#oEF/$\"3Kh3kpbBlWFW7$$\"3QmmmEpS1FF/$\"3[\\kUgu5!*\\FW7$$ \"3%)***\\i`A3v#F/$\"3*H#)3%4[MPcFW7$$\"3Ymmmwy8!z#F/$\"3a#3L9kyJC'FW7 $$\"3/++DOIFLGF/$\"3>RHTH&RM%pFW7$$\"3!****\\(3zMuGF/$\"3(>)\\NbE%[k(F W7$$\"3emm;H_?uep1$F/7$$\"3%*******p(G** y$F/$\"3c7Pn,]%Q?$F/7$$\"3Umm;9@BMQF/$\"3/>.k[uSkLF/7$$\"3/LLL`v&Q(QF/ $\"3,&Q%=I@M6NF/7$$\"30++DOl5;RF/$\"3!G(*f#eUYrOF/7$$\"3/++v.UacRF/$\" 3`@xA@_1GQF/7$$\"\"%F*Ffz-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKN ESSG6#F)-F$6%7$7$F+F+F'FhzF_[l-%*AXESTICKSG6$Fgz\"\"$-%+AXESLABELSG6%% \"tG%%f(t)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F+FfzFf\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)]=L*[u[2](t)*(t-2)^2]" "6#/*&%\"LG\"\"\"7 #-%\"fG6#%\"tGF&*&F%F&7#*&-&%\"uG6#\"\"#6#F+F&*$,&F+F&F3!\"\"F3F&F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=exp(-2*s)*L*[(``(t+2)-2)^2]" "6#/%!G *(-%$expG6#,$*&\"\"#\"\"\"%\"sGF,!\"\"F,%\"LGF,7#*$,&-F$6#,&%\"tGF,F+F ,F,F+F.F+F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=exp(-2*s)*L*[t^2]" "6#/%!G*(-%$expG6#,$*&\"\"#\"\"\" %\"sGF,!\"\"F,%\"LGF,7#*$%\"tGF+F," }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*exp(-2*s)/s^3" "6#/%!G*(\"\"#\" \"\"-%$expG6#,$*&F&F'%\"sGF'!\"\"F'*$F-\"\"$F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f := t -> piecewise(t<2,0,t>=2,(t-2)^2):\n'f(t)'=f(t);\nsimplify( convert(rhs(%),Heaviside)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplac e transform`=inttrans[laplace](f(t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'\"\"#7$*$),&F.! \"\"F'\"\"\"F.F41F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"t G*&-%*HeavisideG6#,&\"\"#!\"\"F'\"\"\"F/,(\"\"%F/*&F1F/F'F/F.*$)F'F-F/ F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"\"# \"\"\"-%$expG6#,$*&F'F(%\"sGF(!\"\"F(F.!\"$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The procedure from the fi rst section which implements the step function " }{XPPEDIT 18 0 "u[a]( t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 274 7 "u[a](t)" }{TEXT -1 19 " may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "u := proc(t)\n local a;\n if t ype(procname,specindex(algebraic,u)) and type(t,algebraic) then\n \+ a := op(1,procname); \n piecewise(t=a,1);\n else 'proc name'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "f := t -> 'u[2](t)'*(t-2)^2 :\n'f(t)'=f(t);\nsimplify(convert(rhs(%),Heaviside)):\nf := unapply(%, t):\n'f(t)'=f(t);\n`Laplace transform`=inttrans[laplace](f(t),t,s);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG*&-&%\"uG6#\"\"#F&\"\" \"),&F'F.F-!\"\"F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG *&-%*HeavisideG6#,&F'\"\"\"\"\"#!\"\"F-,(*$)F'F.F-F-*&\"\"%F-F'F-F/F4F -F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"\"# \"\"\"-%$expG6#,$*&F'F(%\"sGF(!\"\"F(F.!\"$F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplace transform of the function " }{XPPEDIT 18 0 "f( t) = u[Pi/2](t)*sin*t;" "6#/-%\"fG6#%\"tG*(-&%\"uG6#*&%#PiG\"\"\"\"\"# !\"\"6#F'F/%$sinGF/F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "Note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f (t) = PIECEWISE([0, t < Pi/2],[sin*t, Pi/2 <= t]);" "6#/-%\"fG6#%\"tG- %*PIECEWISEG6$7$\"\"!2F'*&%#PiG\"\"\"\"\"#!\"\"7$*&%$sinGF0F'F01*&F/F0 F1F2F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 391 229 229 {PLOTDATA 2 "6--%'CURVESG6%7]s7$$\"31++%*Hjzq:!# <$\"\"\"\"\"!7$$\"3!)[,nQH/0;F*$\"3eD4*4UOT***!#=7$$\"3v(H+ua*GR;F*$\" 3SfzUiDbw**F37$$\"3[Y/8ch`t;F*$\"3K5h\"y*>pZ%=F*$ \"3k\\C*4.Oqi*F37$$\"3;8JMk.'R'>F*$\"3nzN`<],P#*F37$$\"3uN]!*G::$3#F*$ \"3kz)43KBfr)F37$$\"3=s'p#fW>%[jq7F37$$\"3R4[d4*)oQJF *$\"3N9$)QM*RP!H!#?7$$\"3IkgR`dgnKF*$!3'Hck\\k)zc7F37$$\"3=>t@(fAlR$F* $!3\"Gk^.(H2'z$F*$!3#4m?nGqu3'F37$$\"3u?=;y8(*GRF*$!3 %o2?lhe]3(F37$$\"3eUK?CPnlSF*$!31\"QPf#Qn!)zF37$$\"3UkYCqgP-UF*$!3()fS O^?QF()F37$$\"3<(eY5d$yAVF*$!3[VWF*$!3L\">,L HX)R'*F37$$\"3/!\\!eNi'4^%F*$!3e/)=DIHyz*F37$$\"3#>Z7$*RT(yXF*$!3!\\Bd VZC3\"**F37$$\"3(GYy6)*GEh%F*$!3%yE2s'4G]**F37$$\"3#QXWIc;lk%F*$!3Ug'= s'>Jy**F37$$\"3wW/\"\\9//o%F*$!31FEc(G&)[***F37$$\"3rNkxE%4VX=$[ZF*$!3Ep&f)ega$***F37$$\"3a/C%==XBy%F*$!3Q!G 7piSb(**F37$$\"3R*Qv$4>P;[F*$!3%Gm\\pW')f%**F37$$\"3Ot$3pj)R][F*$!3)GM PSt<\\!**F37$$\"3ITV(>4_%=\\F*$!3GD$Ho;S%)y*F37$$\"395./Zb]')\\F*$!3%[ .7l:Zmi*F37$$\"3T(e\\\\xpw6&F*$!3%pksgA<**=*F37$$\"3qk)eG+M)[_F*$!3'oM >h**4`f)F37$$\"3E(3evoVzO&F*$!3+D`16!*4FzF37$$\"3#)4tDsL0([&F*$!3M@k>U #el9(F37$$\"3aT+!zI'oGcF*$!3EPp7$\\=w3'F37$$\"3EtFaV#>.x&F*$!3]S:N(zln !\\F37$$\"3%3#\\0=4I!*eF*$!3#o1<`FY&GQF37$$\"3anqc#f#G5gF*$!3e%f&=4'y_ p#F37$$\"3+*>cJ!)\\)\\hF*$!3=1gXSxSH8F37$$\"3YI`u8qT*G'F*$\"3L:q'>Bm;B 'Fip7$$\"3')\\/(QkdHT'F*$\"37:?+qR3%H\"F37$$\"3Epb*RF)\\OlF*$\"3i[hi.d 71DF37$$\"37#Qz`#>/smF*$\"35fR`EkI\"z$F37$$\"3)\\>jnd&e2oF*$\"3#R*zzM& Rp+&F37$$\"3jn>;'=bm$pF*$\"3a:1(Rr]%zgF37$$\"3=T2c&zCd1(F*$\"3_o.1i]#3 0(F37$$\"33&>xER&R+sF*$\"3f_je%R$3RzF37$$\"3**[Oz*)f1NtF*$\"3MP;QUYd$o )F37$$\"3wGrbPcteuF*$\"3S+/,](=!H#*F37$$\"3a31K&G0Ce(F*$\"3*\\fRPP#\\L '*F37$$\"3W6;Z'\\-\"\\wF*$\"3S))y.5e&3z*F37$$\"3C:Ei2(*z:xF*$\"3Wg#p?f !o/**F37$$\"39<\")>8$[\"\\xF*$\"3=-Z6\"=+^%**F37$$\"3/>Ox=p\\#y(F*$\"3 =$[EfbgW(**F37$$\"3%47\\V_Xe\"yF*$\"37r=*p1HF***F37$$\"3%GiC*HT>\\yF*$ \"3'>vV#*R&))****F37$$\"33>FmeR$Q)yF*$\"3iHsWCXa&***F37$$\"3J:3S(yt%=z F*$\"3eQh&QF37$$\" 3Y)*4pTdcg\"*F*$\"37\"*3)\\0*[6EF37$$\"3e*o(H=V&zG*F*$\"3Q].df8(RO\"F3 7$$\"3q!Q/\\*GM:%*F*$\"3LN\"f*ze#\\V*Fip7$$\"3#)o$o+>'yc&*F*$!3z;w@N;D ;8F37$$\"3seBB&[H#)p*F*$!3e=dA,Kc+FF37$$\"3.&*=)odA`#)*F*$!3e5v*=4*>** QF37$$\"3MJ9`ocT_**F*$!3`)*H5,t$\\.&F37$$\"3gbW8n9\")35!#;$!3G\"R!3%4% \\dhF37$$\"31odTn8QA5Fafl$!3_nDND$Ro;(F37$$\"3\"RQaRrxY.\"Fafl$!3](f;8 s6\"ozF37$$\"3e**H\\gS(p/\"Fafl$!3')Gl\"Q)y0\\')F37$$\"3#=W]^@pd5\"F afl$!3!\\^Qq<82)**F37$$\"39e09[Y246Fafl$!31JdwTaua**F37$$\"3Du$H.5!Q76 Fafl$!3n&o\"zF@!z\"**F37$$\"3Y1qq/5**=6Fafl$!32!zBpgh<\")*F37$$\"3\\QY 34>gD6Fafl$!3g%G\\EKbFm*F37$$\"3U@&pQkG&Q6Fafl$!39SaV$e%=]#*F37$$\"3O/ Wly`X^6Fafl$!3eK9%R%zD$o)F37$$\"3K[n)>Z))\\;\"Fafl$!3LW?cgznMzF37$$\"3 Y#4>`c@&y6Fafl$!3bIo$[F+5/(F37$$\"3INWkUbb\">\"Fafl$!3Q2FtC;NegF37$$\" 3My(p*>&*e/7Fafl$!3m!\\ujbEH(\\F37$$\"3%))*RKW'=z@\"Fafl$!3c[V'oDDex$F 37$$\"3:>#y'oxCJ7Fafl$!3\"HJm#4)R<^#F37$$\"3'p0`6_mWC\"Fafl$!36ymQ5<09 7F37$$\"3'\\*yit_od7Fafl$\"3_=$=n*H>[5!#>7$$\"3?ZdIM>$)p7Fafl$\"3`Zbp. =m:8F37$$\"3W*f$)\\fy>G\"Fafl$\"3$Hkm'3n62DF37$$\"3[s.()Q+!fH\"Fafl$\" 3Zq0+)*R>EQF37$$\"3OXrv#[@)48Fafl$\"3Er#=\"3oBr]F37$$\"3d7#G\"))HFA8Fa fl$\"3!oM72WmB5'F37$$\"3yz#*\\$\\CZL\"Fafl$\"3q(*)**o;2!RqF37$$\"3q)pi _R+![8Fafl$\"3>_DI&p%G)[.K &\\+%fy**F37$$\"3\\OJ))=ie59Fafl$\"3GsMbi/5&***F37$$\"3%*****e)****RT \"Fafl$\"3!*yTJp)f*****F3-%'COLOURG6&%$RGBG$\"*++++\"!\")$F-F-Fcdm-%*T HICKNESSG6#\"\"#-F$6%7$7$FcdmFcdm7$$\"3c'*[zEjzq:F*FcdmF\\dmFddm-F$6&7 #F\\em-%'SYMBOLG6#%'CIRCLEGF\\dm-%&STYLEG6#%&POINTG-F$6&7#7$F]emF+Fbem F\\dmFfem-F$6&F\\fm-Fcem6#%(DIAMONDGF\\dmFfem-F$6&F\\fm-Fcem6#%&CROSSG F\\dmFfem-%%TEXTG6&7$$\"#:F-$!#:!\"#Q\"t6\"-%&COLORG6&F_dm$F,F`gmFfgmF fgm-%%FONTG6$%*HELVETICAG\"#5-%*AXESTICKSG6$7,/F-%\"0G/$\"$d\"F`gm%$p/ 2G/$\"$9$F`gm%\"pG/$\"$r%F`gm%%3p/2G/$\"$G'F`gm%#2pG/$\"%ay!\"$%%5p/2G /$\"%D%*Feim%#3pG/\"#6%%7p/2G/$\"&mD\"Feim%#4pG/$\"%99F`gm%%9p/2G\"\"$ -%+AXESLABELSG6%%!G%%f(t)GFggm-Fhgm6$FcemF[hm-%%VIEWG6$;FcdmF\\gm;$!#7 !\"\"$\"#7Fe[n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = L*[u[Pi/2](t)*sin*t];" "6# /*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F%F&7#*(-&%\"uG6#*&%#PiGF&\"\"#!\"\"6 #F+F&%$sinGF&F+F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = exp(-Pi*s/2)*L*[sin(t+Pi/2)];" "6#/%!G*(-%$expG 6#,$*(%#PiG\"\"\"%\"sGF,\"\"#!\"\"F/F,%\"LGF,7#-%$sinG6#,&%\"tGF,*&F+F ,F.F/F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(-Pi*s/2)*L*[cos*t];" "6#/%!G*(-%$expG6#,$*(%#P iG\"\"\"%\"sGF,\"\"#!\"\"F/F,%\"LGF,7#*&%$cosGF,%\"tGF,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-Pi* s/2)*s/(s^2+1);" "6#/%!G*(-%$expG6#,$*(%#PiG\"\"\"%\"sGF,\"\"#!\"\"F/F ,F-F,,&*$F-F.F,F,F,F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "f := t -> piecewise(t

=Pi/2,sin(t)):\n'f(t)'=f(t);\nsimplify(convert(rhs(%),Heavisid e)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=inttrans[l aplace](f(t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-% *PIECEWISEG6$7$\"\"!2F',$*&\"\"#!\"\"%#PiG\"\"\"F37$-%$sinGF&1F.F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG*&-%$sinGF&\"\"\"-%*Heav isideG6#,&*&\"\"#!\"\"%#PiGF+F2F'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*(-%$expG6#,$*(\"\"#!\"\"%\"sG\"\"\"%#PiGF.F,F .F-F.,&*$)F-F+F.F.F.F.F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 72 "The procedure from the first section which impleme nts the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"a G6#%\"tG%!G" }{TEXT 274 7 "u[a](t)" }{TEXT -1 19 " may also be used. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "u := proc(t)\n local a;\n if type(procname,specindex(alge braic,u)) and type(t,algebraic) then\n a := op(1,procname); \n \+ piecewise(t=a,1);\n else 'procname'(t)\n end if;\nend p roc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f := t -> 'u[Pi/2](t)'*sin(t):\n'f(t)'=f(t);\nsimpli fy(convert(rhs(%),Heaviside)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Lap lace transform`=inttrans[laplace](f(t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG*&-&%\"uG6#,$*&\"\"#!\"\"%#PiG\"\"\"F2F&F 2-%$sinGF&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG*&-%*Hea visideG6#,&*&\"\"#!\"\"%#PiG\"\"\"F/F'F1F1-%$sinGF&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG*(-%$expG6#,$*(\"\"#!\"\"%\"s G\"\"\"%#PiGF.F,F.F-F.,&*$)F-F+F.F.F.F.F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 46 " We find the Laplace transform of the function " }{XPPEDIT 18 0 "f(t) = u[Pi](t)*24*exp(-t)*cos*2*t;" "6#/-%\"fG6#%\"tG*.-&%\"uG6#%#PiG6#F'\" \"\"\"#CF/-%$expG6#,$F'!\"\"F/%$cosGF/\"\"#F/F'F/" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 12 "Note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([0, t < Pi],[24*exp(-t)*cos* 2*t, Pi <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'%#PiG7$*,\" #C\"\"\"-%$expG6#,$F'!\"\"F2%$cosGF2\"\"#F2F'F21F.F'" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 411 221 221 {PLOTDATA 2 "6,-%'CURVESG6%7hn7$$\"3))***Hnl#fTJ!#<$\"3g=P2NS8P5F*7$$\"3)=nvNQUX >$F*$\"3OEpWG9O\"y*!#=7$$\"3MW8U5@\\ZKF*$\"3GQH.4/r?\"*F27$$\"3!G6HYQj NH$F*$\"3NED$>tk2])F27$$\"3E\")o$)eYjRLF*$\"31c))\\'\\!G\\yF27$$\"3-ow 7DmW\"R$F*$\"3s'R>h\"He!4(F27$$\"3Db%=9feKW$F*$\"3L)eO*za5+9a\\$F*$\"3ee')y8dcKbF27$$\"3qI**o5%pva$F*$\"31d!f\\)H7bZF27$$\"3/ DywMpZ*f$F*$\"3)*pm9JS\"p*RF27$$\"3#)>d%)eWQ^OF*$\"3?%e-tW#QiKF27$$\"3 /=:Hy$4&*p$F*$\"32`+Tgcp3EF27$$\"3G;tt(HMwu$F*$\"3!efl*=.A')>F27$$\"3= <$>V'[Y(z$F*$\"3EaT(4&H$)y8F27$$\"35=8!4V&HZQF*$\"3vbtijI$y7)!#>7$$\"3 S_iQ<^O]RF*$!3W:=Hgxte@F^p7$$\"3Sf%z!eU5`SF*$!3ss..cDFS5F27$$\"3[`!GD4 %yeTF*$!3QPm7'=7qn\"F27$$\"3_4JB[p'=D%F*$!3_&HNUfms1#F27$$\"3[\\yos:E/ VF*$!3G!\\ooCQ5A#F27$$\"3W*eUr>cmN%F*$!3&\\'3/0>gIBF27$$\"3R7$f*efE4WF *$!3U#o\\rTH\"*R#F27$$\"3MNgx?d(=Y%F*$!3L0y\"4$[LHCF27$$\"3Ib#QY(\\d7X F*$!3L>w?\"[*QDCF27$$\"3Ev/]GUFjXF*$!3#Q,T[&=#zvodF*$\"3)3_v IX7o'QF^p7$$\"3u!3R;Gk%peF*$\"3?Ro!oEVue%F^p7$$\"3A=1U0Q]tfF*$\"3+gH5D -.u\\F^p7$$\"3SE;)>='GvgF*$\"3S3M%GRUu/&F^p7$$\"3)fpc['ewthF*$\"3av?q) o\\3)[F^p7$$\"3YrJOi26$G'F*$\"3c>\"f$=m>#[%F^p7$$\"3;BN3;?O\"Q'F*$\"3g x3\")f**o%)RF^p7$$\"3+,-u+iE'['F*$\"3U5]Nk%Q0O$F^p7$$\"3'ppmQ)\\K\"e'F *$\"3#47:Jp)Q_FF^p7$$\"3cc)Q@M\\_o'F*$\"3kk%QT'=*)z?F^p7$$\"3WM$eU1MIy 'F*$\"3m:_p/\\mp9F^p7$$\"3#H:gUFZ_)oF*$\"3`\\v$H;D,!))Fcv7$$\"3sc2\"*p $z^)pF*$\"3cant5U(zo$Fcv7$$\"3/IB$G,*z*3(F*$!3)G,l1*z-'[)!#@7$$\"3a:`I j,c!>(F*$!3+#Hsl\\\"**oVFcv7$$\"3)zO>H*Hg$H(F*$!3al9?(R'*y4(Fcv7$$\"3% R,zed#z&R(F*$!3YhcN&H'Gn*)Fcv7$$\"3q/%>'*=%p*[(F*$!3s0Sy(em1+\"F^p7$$ \"3'ocdd.;tf(F*$!3cSl-%*=4\\5F^p7$$\"3/!)G\"zHuNp(F*$!3s#fIfe%)z.\"F^p 7$$\"3+'*y;#o0iz(F*$!3#RymvZ&>0)*Fcv7$$\"3!4]pB,PW*yF*$!3@Y>T7*4#=*)Fc v7$$\"3A+++#*******zF*$!3GA'*[!G:-r(Fcv-%'COLOURG6&%$RGBG$\"*++++\"!\" )$\"\"!Fj^lFi^l-%*THICKNESSG6#\"\"#-F$6%7$7$Fi^lFi^l7$$\"37$z*e`EfTJF* Fi^lFb^lF[_l-F$6&7#Fc_l-%'SYMBOLG6#%'CIRCLEGFb^l-%&STYLEG6#%&POINTG-F$ 6&7#7$Fd_l$\"3/+++QS8P5F*Fi_lFb^lF]`l-F$6&Fc`l-Fj_l6#%(DIAMONDGFb^lF]` l-F$6&Fc`l-Fj_l6#%&CROSSGFb^lF]`l-%%TEXTG6&7$$\"#$)!\"\"$!\"#FgalQ\"t6 \"-%&COLORG6&Fe^l$\"\"\"FialF_blF_bl-%%FONTG6$%*HELVETICAG\"#5-%*AXEST ICKSG6$\"\"&F^_l-%+AXESLABELSG6%%!G%%f(t)G-Fbbl6#%(DEFAULTG-%%VIEWG6$; Fi^l$\"\")Fj^l;$!\"$Fgal$\"#6Fgal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = L*[u[Pi](t)*24*exp(-t)*cos*2*t];" "6# /*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F%F&7#*.-&%\"uG6#%#PiG6#F+F&\"#CF&-%$ expG6#,$F+!\"\"F&%$cosGF&\"\"#F&F+F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 24*exp(-Pi*s)*L*[exp(-(t+Pi))*cos(2*(t+Pi))];" "6#/%!G**\"#C\"\" \"-%$expG6#,$*&%#PiGF'%\"sGF'!\"\"F'%\"LGF'7#*&-F)6#,$,&%\"tGF'F-F'F/F '-%$cosG6#*&\"\"#F',&F7F'F-F'F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 24*exp(-Pi*s)*L*[exp(-t-Pi)*cos(2*t+2*Pi)];" "6#/%!G**\"#C\"\"\"-%$ expG6#,$*&%#PiGF'%\"sGF'!\"\"F'%\"LGF'7#*&-F)6#,&%\"tGF/F-F/F'-%$cosG6 #,&*&\"\"#F'F6F'F'*&F " 0 "" {MPLTEXT 1 0 195 "f := t -> piecewise(t=Pi,24*exp(-t)*cos(2*t)):\n'f(t)'=f(t);\nsimplify(convert(rhs(%),He aviside)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=simp lify(inttrans[laplace](f(t),t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'%#PiG7$,$*(\"#C\"\"\"-%$expG6#, $F'!\"\"F3-%$cosG6#,$*&\"\"#F3F'F3F3F3F31F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,$**\"#C\"\"\"-%$expG6#,$F'!\"\"F+-%$cosG 6#,$*&\"\"#F+F'F+F+F+-%*HeavisideG6#,&%#PiGF0F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$**\"#C\"\"\"-%$expG6#,$*&,& %\"sGF(F(F(F(%#PiGF(!\"\"F(F.F(,(*$)F/\"\"#F(F(*&F5F(F/F(F(\"\"&F(F1F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The \+ procedure from the first section which implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 274 7 "u[a](t)" }{TEXT -1 19 " may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "u := proc(t )\n local a;\n if type(procname,specindex(algebraic,u)) and type(t ,algebraic) then\n a := op(1,procname); \n piecewise(t=a,1);\n else 'procname'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "f := t -> 'u[Pi](t)'*24*exp(-t)*cos(2*t):\n'f(t)'=f(t);\nsimplify(convert(rh s(%),Heaviside)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transfor m`=simplify(inttrans[laplace](f(t),t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,$**\"#C\"\"\"-&%\"uG6#%#PiGF&F+-%$expG6# ,$F'!\"\"F+-%$cosG6#,$*&\"\"#F+F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,$**\"#C\"\"\"-%*HeavisideG6#,&%#PiG!\"\" F'F+F+-%$expG6#,$F'F1F+-%$cosG6#,$*&\"\"#F+F'F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$**\"#C\"\"\"-%$expG6#,$*&,& %\"sGF(F(F(F(%#PiGF(!\"\"F(F.F(,(*$)F/\"\"#F(F(*&F5F(F/F(F(\"\"&F(F1F( " }}}{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 49 "Finding Laplace transforms of pi ecewise functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplace trans form of the function " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" } {TEXT -1 24 " defined by the formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([0, t < 2],[10*e xp(-t), 2 <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'\"\"#7$*& \"#5\"\"\"-%$expG6#,$F'!\"\"F21F.F'" }{TEXT -1 3 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 447 242 242 {PLOTDATA 2 "6+-%'CURVESG6% 7S7$$\"\"#\"\"!$\"3-FhOKGN`8!#<7$$\"3Wmmm;')=(3#F-$\"3T5%>.d`.C\"F-7$$ \"3WLL$e'40j@F-$\"3K*[VB/Q(\\6F-7$$\"3!pmm;6m$[AF-$\"3%y'G))edrb5F-7$$ \"3xmmm\"yYUL#F-$\"3K9!yD%HM)o*!#=7$$\"3CLL$eF>(>CF-$\"3yBb/T$eY*))FB7 $$\"3'omm\">K'*)\\#F-$\"3.G/'4p9q@)FB7$$\"3/++]Kd,\"e#F-$\"3;?n&zp2(pv FB7$$\"3Qmm;fX(em#F-$\"39kZ&*Q6&Q&pFB7$$\"37++]U7Y]FF-$\"3[c%Hf1k9o(\\FB7$$\"3)*******\\*3q3$F-$\"3]w;twc#Q c%FB7$$\"3?+++q=\\qJF-$\"3O10SnUH)>%FB7$$\"3mmm;fBIYKF-$\"3!R#**RKYy\" *QFB7$$\"30LLLj$[kL$F-$\"31S0zoSIcNFB7$$\"3?LLL`Q\"GT$F-$\"3#4/znzN[H$ FB7$$\"3o****\\s]k,NF-$\"3m)*>fPZx9IFB7$$\"3#HLLLvv-e$F-$\"3ATp#)\\7!o y#FB7$$\"33++]sgamOF-$\"3es!=kfhkb#FB7$$\"3!)****\\ FB7$$\"3immmTc-)*RF-$\"3)=;_Wi$=N=FB7$$\"3Mmm;f`@'3%F-$\"3qej!zbq-o\"F B7$$\"3M****\\nZ)H;%F-$\"3w'e#=WU5c:FB7$$\"3YmmmJy*eC%F-$\"3')Q]'z$oGK 9FB7$$\"3')******R^bJVF-$\"3))\\@Y-'3ZJ\"FB7$$\"3f*****\\5a`T%F-$\"3M2 q69s-47FB7$$\"3o****\\7RV'\\%F-$\"3tv=C\"Ho[6\"FB7$$\"3k*****\\@fke%F- $\"3+L&=\"=r))=5FB7$$\"3/LLL`4NnYF-$\"3O'y?,-IrR*!#>7$$\"3[******\\,s` ZF-$\"3:r!yERp&>')Fiu7$$\"3[mm;zM)>$[F-$\"33\")HFnepqzFiu7$$\"3$****** *pfa<\\F-$\"3c2S\\H^11iFiu7$$\"3;LLL$)G[k^F-$\"3cT,,Bz,;dFiu7$$\"3# )****\\7yh]_F-$\"3=GG)zTxUC&Fiu7$$\"3Kmmm')fdL`F-$\"3**prz<'yn#[Fiu7$$ \"36mmm,FT=aF-$\"3itCTI%zTV%Fiu7$$\"3FLL$e#pa-bF-$\"3[jYcyfPwSFiu7$$\" 3!*******Rv&)zbF-$\"3%*f'fn*H5tPFiu7$$\"3%GLL$GUYocF-$\"3k%>X!yW;`MFiu 7$$\"33mmm1^rZdF-$\"3!))4'pD91!>$Fiu7$$\"34++]sI@KeF-$\"3OAgW#*>eJHFiu 7$$\"34++]2%)38fF-$\"3`8'H0ZBQq#Fiu7$$\"\"'F*$\"3oejmw@vyCFiu-%'COLOUR G6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-%*THICKNESSG6#F)-F$6%7$7$Fa[lFa[l7$F( Fa[lFjzFb[l-F$6&7#Fi[l-%'SYMBOLG6#%'CIRCLEGFjz-%&STYLEG6#%&POINTG-F$6& 7#7$F($\"37+++KGN`8F-F]\\lFjzFa\\l-F$6&Fg\\l-F^\\l6#%(DIAMONDGFjzFa\\l -F$6&Fg\\l-F^\\l6#%&CROSSGFjzFa\\l-%*AXESTICKSG6$\"\"%\"\"$-%+AXESLABE LSG6%%\"tG%%f(t)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fa[lFfz;Fa[l$\"#>!\"\" " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "By ta king the product of function " }{XPPEDIT 18 0 "g(t) = 10*exp(-t);" "6# /-%\"gG6#%\"tG*&\"#5\"\"\"-%$expG6#,$F'!\"\"F*" }{TEXT -1 24 " with th e step function " }{XPPEDIT 18 0 "u[2](t)" "6#-&%\"uG6#\"\"#6#%\"tG" } {TEXT -1 15 ", the function " }{XPPEDIT 18 0 "g(t);" "6#-%\"gG6#%\"tG " }{TEXT -1 23 " is \"switched on\" when " }{XPPEDIT 18 0 "t = 2" "6#/ %\"tG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The functio n " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 51 " can be des cribed using the unit step function as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = u[2](t)*g(t);" "6#/-%\"fG6#%\"tG*&-&% \"uG6#\"\"#6#F'\"\"\"-%\"gG6#F'F/" }{XPPEDIT 18 0 "``=u[2](t)*10*exp(- t)" "6#/%!G*(-&%\"uG6#\"\"#6#%\"tG\"\"\"\"#5F--%$expG6#,$F,!\"\"F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The Laplace transform o f " }{XPPEDIT 18 0 "g(t) = 10*exp(-t);" "6#/-%\"gG6#%\"tG*&\"#5\"\"\"- %$expG6#,$F'!\"\"F*" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "L*[g(t)] = 10/ (s+1);" "6#/*&%\"LG\"\"\"7#-%\"gG6#%\"tGF&*&\"#5F&,&%\"sGF&F&F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Using the second shift \+ formula we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L *[f(t)] = L*[u[2](t)*g(t)];" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F%F&7 #*&-&%\"uG6#\"\"#6#F+F&-%\"gG6#F+F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=L*[u[2]*10*exp(-t)]" "6#/%!G*&% \"LG\"\"\"7#*(&%\"uG6#\"\"#F'\"#5F'-%$expG6#,$%\"tG!\"\"F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = exp( -2*s)*L*[10*exp(-(t+2))];" "6#/%!G*(-%$expG6#,$*&\"\"#\"\"\"%\"sGF,!\" \"F,%\"LGF,7#*&\"#5F,-F'6#,$,&%\"tGF,F+F,F.F,F," }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = exp(-2*s)*L*[10 *exp(-t-2)];" "6#/%!G*(-%$expG6#,$*&\"\"#\"\"\"%\"sGF,!\"\"F,%\"LGF,7# *&\"#5F,-F'6#,&%\"tGF.F+F.F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(-2*s)*exp(-2)*L*[10*exp(-t)]; " "6#/%!G**-%$expG6#,$*&\"\"#\"\"\"%\"sGF,!\"\"F,-F'6#,$F+F.F,%\"LGF,7 #*&\"#5F,-F'6#,$%\"tGF.F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 10*exp(-2*s-2)/(s+1);" "6#/%!G*( \"#5\"\"\"-%$expG6#,&*&\"\"#F'%\"sGF'!\"\"F-F/F',&F.F'F'F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" }{TEXT -1 87 " from the subsection \+ which follows the examples and which implements the step function " } {XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 274 7 "u[a](t)" }{TEXT -1 46 " may be used to find the Laplace transform o f " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f := t -> 'u[2](t)'*10*exp(-t):\n'f(t)'=f(t);\nsimplify(convert(rhs(%), Heaviside)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=si mplify(inttrans[laplace](f(t),t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"fG6#%\"tG,$*(\"#5\"\"\"-&%\"uG6#\"\"#F&F+-%$expG6#,$F'!\"\"F+F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,$*(\"#5\"\"\"-%*Hea visideG6#,&\"\"#!\"\"F'F+F+-%$expG6#,$F'F1F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"#5\"\"\"-%$expG6#,&*&\"\"#F( %\"sGF(!\"\"F.F0F(,&F/F(F(F(F0F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can start with the piecewise definition of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 2 ": " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([0, t < 2],[10*exp(- t), 2 <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'\"\"#7$*&\"#5 \"\"\"-%$expG6#,$F'!\"\"F21F.F'" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f := t -> p iecewise(t<2,0,t>=2,10*exp(-t)):\n'f(t)'=f(t);\nsimplify(convert(f(t), Heaviside)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=in ttrans[laplace](f(t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"tG-%*PIECEWISEG6$7$\"\"!2F'\"\"#7$,$*&\"#5\"\"\"-%$expG6#,$F'!\"\" F3F31F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,$*(\"#5\" \"\"-%*HeavisideG6#,&F'F+\"\"#!\"\"F+-%$expG6#,$F'F1F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,$*(\"#5\"\"\"-%$expG6#,&\" \"#!\"\"*&F-F(%\"sGF(F.F(,&F0F(F(F(F.F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplace transform of the func tion " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 25 " defined by the formula: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {XPPEDIT 18 0 "f(t) = PIECEWISE([t, t < 2],[10*exp(-t), 2 <= t]);" " 6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$F'2F'\"\"#7$*&\"#5\"\"\"-%$expG6#,$F' !\"\"F11F-F'" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 442 204 204 {PLOTDATA 2 "6 +-%'CURVESG6%7S7$$\"\"#\"\"!$\"3-FhOKGN`8!#<7$$\"3Wmmm;')=(3#F-$\"3T5% >.d`.C\"F-7$$\"3WLL$e'40j@F-$\"3K*[VB/Q(\\6F-7$$\"3!pmm;6m$[AF-$\"3%y' G))edrb5F-7$$\"3xmmm\"yYUL#F-$\"3K9!yD%HM)o*!#=7$$\"3CLL$eF>(>CF-$\"3y Bb/T$eY*))FB7$$\"3'omm\">K'*)\\#F-$\"3.G/'4p9q@)FB7$$\"3/++]Kd,\"e#F-$ \"3;?n&zp2(pvFB7$$\"3Qmm;fX(em#F-$\"39kZ&*Q6&Q&pFB7$$\"37++]U7Y]FF-$\" 3[c%Hf1k9o(\\FB7$$\"3)*******\\*3q3$F -$\"3]w;twc#Qc%FB7$$\"3?+++q=\\qJF-$\"3O10SnUH)>%FB7$$\"3mmm;fBIYKF-$ \"3!R#**RKYy\"*QFB7$$\"30LLLj$[kL$F-$\"31S0zoSIcNFB7$$\"3?LLL`Q\"GT$F- $\"3#4/znzN[H$FB7$$\"3o****\\s]k,NF-$\"3m)*>fPZx9IFB7$$\"3#HLLLvv-e$F- $\"3ATp#)\\7!oy#FB7$$\"33++]sgamOF-$\"3es!=kfhkb#FB7$$\"3!)****\\FB7$$\"3immmTc-)*RF-$\"3)=;_Wi$=N=FB7$$\"3Mmm;f`@'3%F-$ \"3qej!zbq-o\"FB7$$\"3M****\\nZ)H;%F-$\"3w'e#=WU5c:FB7$$\"3YmmmJy*eC%F -$\"3')Q]'z$oGK9FB7$$\"3')******R^bJVF-$\"3))\\@Y-'3ZJ\"FB7$$\"3f***** \\5a`T%F-$\"3M2q69s-47FB7$$\"3o****\\7RV'\\%F-$\"3tv=C\"Ho[6\"FB7$$\"3 k*****\\@fke%F-$\"3+L&=\"=r))=5FB7$$\"3/LLL`4NnYF-$\"3O'y?,-IrR*!#>7$$ \"3[******\\,s`ZF-$\"3:r!yERp&>')Fiu7$$\"3[mm;zM)>$[F-$\"33\")HFnepqzF iu7$$\"3$*******pfa<\\F-$\"3c2S\\H^11iFiu7$$\"3;LLL$)G[k^F-$\"3cT,, Bz,;dFiu7$$\"3#)****\\7yh]_F-$\"3=GG)zTxUC&Fiu7$$\"3Kmmm')fdL`F-$\"3** prz<'yn#[Fiu7$$\"36mmm,FT=aF-$\"3itCTI%zTV%Fiu7$$\"3FLL$e#pa-bF-$\"3[j YcyfPwSFiu7$$\"3!*******Rv&)zbF-$\"3%*f'fn*H5tPFiu7$$\"3%GLL$GUYocF-$ \"3k%>X!yW;`MFiu7$$\"33mmm1^rZdF-$\"3!))4'pD91!>$Fiu7$$\"34++]sI@KeF-$ \"3OAgW#*>eJHFiu7$$\"34++]2%)38fF-$\"3`8'H0ZBQq#Fiu7$$\"\"'F*$\"3oejmw @vyCFiu-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-%*THICKNESSG6#F)-F$6% 7S7$Fa[lFa[l7$$\"39LLLL3VfVFiuFj[l7$$\"3'pmm;H[D:)FiuF]\\l7$$\"3LLLLe0 $=C\"FBF`\\l7$$\"3ILLL3RBr;FBFc\\l7$$\"3Ymm;zjf)4#FBFf\\l7$$\"3=LL$e4; [\\#FBFi\\l7$$\"3p****\\i'y]!HFBF\\]l7$$\"3,LL$ezs$HLFBF_]l7$$\"3_**** \\7iI_PFBFb]l7$$\"3#pmmm@Xt=%FBFe]l7$$\"3QLLL3y_qXFBFh]l7$$\"3i****** \\1!>+&FBF[^l7$$\"3()******\\Z/NaFBF^^l7$$\"3'*******\\$fC&eFBFa^l7$$ \"3ELL$ez6:B'FBFd^l7$$\"3Smmm;=C#o'FBFg^l7$$\"3-mmmm#pS1(FBFj^l7$$\"3] ****\\i`A3vFBF]_l7$$\"3slmmm(y8!zFBF`_l7$$\"3V++]i.tK$)FBFc_l7$$\"39++ ](3zMu)FBFf_l7$$\"3#pmm;H_?<*FBFi_l7$$\"3emm;zihl&*FBF\\`l7$$\"39LLL3# G,***FBF_`l7$$\"3F-Fadl7$$\"3/++v.U ac>F-Fddl7$F(F(FjzFb[l-F$6&7#Ffdl-%'SYMBOLG6#%'CIRCLEGFjz-%&STYLEG6#%& POINTG-F$6&7#7$F($\"37+++KGN`8F-FjdlFjzF^el-F$6&Fdel-F[el6#%(DIAMONDGF jzF^el-F$6&Fdel-F[el6#%&CROSSGFjzF^el-%*AXESTICKSG6$\"\"%\"\"$-%+AXESL ABELSG6%%\"tG%%f(t)G-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fa[lFfz;Fa[l$\"#A! \"\"" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 44.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "At " } {XPPEDIT 18 0 "t = 2" "6#/%\"tG\"\"#" }{TEXT -1 21 " we switch over fr om " }{XPPEDIT 18 0 "f(t) = t;" "6#/-%\"fG6#%\"tGF'" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "f(t) = 10*exp(-t);" "6#/-%\"fG6#%\"tG*&\"#5\"\"\"-%$ expG6#,$F'!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The step function " }{XPPEDIT 18 0 "u[2](t);" "6#-&%\"uG6#\"\"#6#%\"tG" } {TEXT -1 19 " can be used to do " }{TEXT 259 10 "two things" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t;" "6#/-%\"fG6#%\"tGF'" } {TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 259 9 "switch on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 10*exp(-t);" "6#/-%\"fG6#% \"tG*&\"#5\"\"\"-%$expG6#,$F'!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "To do this form a facto r with the function to be switched off " }{TEXT 259 10 "subtracted" } {TEXT -1 36 " and the function to be switched on " }{TEXT 259 5 "added " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus the function " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\" tG" }{TEXT -1 50 " can be described using the unit step function as:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t+u[2](t)*(- t+10*exp(-t));" "6#/-%\"fG6#%\"tG,&F'\"\"\"*&-&%\"uG6#\"\"#6#F'F),&F'! \"\"*&\"#5F)-%$expG6#,$F'F2F)F)F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 39 "Using the second shift formula we have:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = L*[t+u[2](t)*(-t+10*e xp(-t))];" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F%F&7#,&F+F&*&-&%\"uG6# \"\"#6#F+F&,&F+!\"\"*&\"#5F&-%$expG6#,$F+F7F&F&F&F&F&" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "L*[t]+L*[u[2](t )*(-t+10*exp(-t))];" "6#,&*&%\"LG\"\"\"7#%\"tGF&F&*&F%F&7#*&-&%\"uG6# \"\"#6#F(F&,&F(!\"\"*&\"#5F&-%$expG6#,$F(F3F&F&F&F&F&" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(s^2);" "6#*& \"\"\"F$*$%\"sG\"\"#!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "exp(-2*s) *L*[-t-2+10*exp(-t-2)];" "6#*(-%$expG6#,$*&\"\"#\"\"\"%\"sGF*!\"\"F*% \"LGF*7#,(%\"tGF,F)F,*&\"#5F*-F%6#,&F0F,F)F,F*F*F*" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(s^2);" "6#*&\"\" \"F$*$%\"sG\"\"#!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "exp(-2*s)*L*[ -t-2];" "6#*(-%$expG6#,$*&\"\"#\"\"\"%\"sGF*!\"\"F*%\"LGF*7#,&%\"tGF,F )F,F*" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "exp(-2*s)*exp(-2)*L*[10*exp( -t)];" "6#**-%$expG6#,$*&\"\"#\"\"\"%\"sGF*!\"\"F*-F%6#,$F)F,F*%\"LGF* 7#*&\"#5F*-F%6#,$%\"tGF,F*F*" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "1/(s^2);" "6#*&\"\"\"F$*$%\"sG\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-exp(-2*s)/(s^2);" "6#,$*&-%$expG6#,$*&\"\"#\"\"\"% \"sGF+!\"\"F+*$F,F*F-F-" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-2*exp(-2*s) /s;" "6#,$*(\"\"#\"\"\"-%$expG6#,$*&F%F&%\"sGF&!\"\"F&F,F-F-" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "10*exp(-2*(s+1))/(s+1);" "6#*(\"#5\"\"\"-%$ expG6#,$*&\"\"#F%,&%\"sGF%F%F%F%!\"\"F%,&F-F%F%F%F." }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-2*exp(-2*s)/s+(1-exp(-2*s))/s^2+10*exp(-2-2*s)/(1+s )" "6#/%!G,(*(\"\"#\"\"\"-%$expG6#,$*&F'F(%\"sGF(!\"\"F(F.F/F/*&,&F(F( -F*6#,$*&F'F(F.F(F/F/F(*$F.F'F/F(*(\"#5F(-F*6#,&F'F/*&F'F(F.F(F/F(,&F( F(F.F(F/F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 1 "u" }{TEXT -1 87 " f rom the subsection which follows the examples and which implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"t G%!G" }{TEXT 274 7 "u[a](t)" }{TEXT -1 46 " may be used to find the La place transform of " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 19 "c ollect_like_denoms" }{TEXT -1 85 " also from the subsection which foll ows the examples is used to simplify the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "f := t -> t +'u[2](t)'*(-t+10*exp(-t)):\n'f(t)'=f(t);\nconvert(rhs(%),Heaviside): \nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=inttrans[lapla ce](f(t),t,s);\n``=collect_like_denoms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,&F'\"\"\"*&-&%\"uG6#\"\"#F&F),&F'!\"\"*& \"#5F)-%$expG6#,$F'F1F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" fG6#%\"tG,(F'\"\"\"*&-%*HeavisideG6#,&\"\"#!\"\"F'F)F)F'F)F0*(\"#5F)F+ F)-%$expG6#,$F'F0F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tr ansformG,**&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*(F+F'-%$expG6#,$*&F+F'F*F'F ,F'F*F,F,*&F.F'F*!\"#F,*(\"#5F'-F/6#,&*&F+F'F*F'F,F+F,F',&F*F'F'F'F,F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#\"\"\"-%$expG6#,$*&F' F(%\"sGF(!\"\"F(F.F/F/*&,&F(F/F)F(F(F.!\"#F/*(\"#5F(-F*6#,&*&F'F(F.F(F /F'F/F(,&F.F(F(F(F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Altern atively, we can start with the piecewise definition of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([t, t < 2],[10*exp(-t), 2 <= t]);" "6# /-%\"fG6#%\"tG-%*PIECEWISEG6$7$F'2F'\"\"#7$*&\"#5\"\"\"-%$expG6#,$F'! \"\"F11F-F'" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "f := t -> piecewise(t<2,t,t >=2,10*exp(-t)):\n'f(t)'=f(t);\nsimplify(convert(f(t),Heaviside)):\nf \+ := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=inttrans[laplace]( f(t),t,s);\n``=collect_like_denoms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$F'2F'\"\"#7$,$*&\"#5\"\" \"-%$expG6#,$F'!\"\"F2F21F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" fG6#%\"tG,(F'\"\"\"*&-%*HeavisideG6#,&\"\"#!\"\"F'F)F)F'F)F0*(\"#5F)F+ F)-%$expG6#,$F'F0F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~tr ansformG,**&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*(F+F'-%$expG6#,$*&F+F'F*F'F ,F'F*F,F,*&F.F'F*!\"#F,*(\"#5F'-F/6#,&*&F+F'F*F'F,F+F,F',&F*F'F'F'F,F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#\"\"\"-%$expG6#,$*&F' F(%\"sGF(!\"\"F(F.F/F/*&,&F(F/F)F(F(F.!\"#F/*(\"#5F(-F*6#,&*&F'F(F.F(F /F'F/F(,&F.F(F(F(F/F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplace transform of the function " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 25 " defined by the \+ formula: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([2*t, t < 1],[2, 1 <= t and t < 3],[1, 3 <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$*&\"\"#\"\"\"F'F.2F'F.7$ F-31F.F'2F'\"\"$7$F.1F4F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This function has a \"jump\" disco ntinuity where " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"\"\"" }{TEXT -1 37 ", where the value jumps from 2 to 1. " }}{PARA 0 "" 0 "" {TEXT -1 145 " Such \"jump\" discontinuities are often shown (incorrectly) on a graph with a vertical line joining the two points associated with the two v alues. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "f := t -> piecewise(t<1,2*t,t<3,2,t>=3,1):\n'f(t)'=f (t);\nplot(f(t),t=0..5,y=0..2.5,thickness=2,color=red,\n \+ labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$,$*&\"\"#\"\"\"F'F/F/2F'F/7$F.2F'\" \"$7$F/1F3F'" }}{PARA 13 "" 1 "" {GLPLOT2D 358 192 192 {PLOTDATA 2 "6( -%'CURVESG6#7[p7$$\"\"!F)F(7$$\"3WmmmT&)G\\a!#>$\"3GLLL3x&)*3\"!#=7$F. $\"3emmm;arz@F07$$\"3))**\\i!R(*Rc\"F0$\"3v***\\7y%*z7$F07$$\"3umm\"H2 P\"Q?F0$\"3[LL$e9ui2%F07$$\"3!***\\PMnNrDF0$\"3z***\\(oMrU^F07$$\"3MLL $eRwX5$F0$\"3nmmm\"z_\"4iF07$$\"3rLLL$eI8k$F0$\"3Unmmm6m#G(F07$$\"33ML $3x%3yTF0$\"39ommT&phN)F07$$\"3h+]PfyG7ZF0$\"3A,+v=ddC%*F07$$\"3emm\"z %4\\Y_F0$\"3KLLe*=)H\\5!#<7$$\"32++v$flMLe*)>VB$)F0$\"3%om;zR'ok;Ffn7$$\"3wmmTg()4_))F0$\"3OLL3_(>/x\" Ffn7$$\"3Y++DJbw!Q*F0$\"33++D1J:w=Ffn7$$\"3=nT&)3\\m_'*F0$\"3VL3x\")H` I>Ffn7$$\"3+N$ekGkX#**F0$\"3+n;HdG\"\\)>Ffn7$$\"3)f&)e'3Ffn7$$\"3%ePf38RD***F0$\"3;v=Ffn7$$\"3d*)fIbEl-5Ffn$\" \"#F)7$$\"3nTg_(R^g+\"FfnFfr7$$\"3'e9m>))[G,\"FfnFfr7$$\"31]iSmjk>5Ffn Ffr7$$\"3XekGN8CL5FfnFfr7$$\"3%ommTIOo/\"FfnFfr7$$\"3E+]7GTt%4\"FfnFfr 7$$\"3YLL3_>jU6FfnFfr7$$\"37++]i^Z]7FfnFfr7$$\"33++](=h(e8FfnFfr7$$\"3 /++]P[6j9FfnFfr7$$\"3UL$e*[z(yb\"FfnFfr7$$\"3wmm;a/cq;FfnFfr7$$\"3%omm mJFfnFfr7$$\"3K+]i!f# =$3#FfnFfr7$$\"3?+](=xpe=#FfnFfr7$$\"37nm\"H28IH#FfnFfr7$$\"3um;zpSS\" R#FfnFfr7$$\"3GLL3_?`(\\#FfnFfr7$$\"3fL$e*)>pxg#FfnFfr7$$\"33+]Pf4t.FF fnFfr7$$\"3uLLe*Gst!GFfnFfr7$$\"3)om\"H2\"34'GFfnFfr7$$\"30+++DRW9HFfn Ffr7$$\"3S+Dc,6jSHFfnFfr7$$\"3K+]7y#=o'HFfnFfr7$$\"3G]iSm=\"*zHFfnFfr7 $$\"3C+voaa+$*HFfnFfr7$$\"3%H\"yv^)yi*HFfnFfr7$$\"3AD\"G)[Ab**HFfnFfr7 $$\"3[P%)*ekDG+$Ffn$\"\"\"F)7$$\"3>](oH/*41IFfnFhx7$$\"3>IFfnFhx7$$\"3A+v$4^n)pIFfnFhx7$$\"3F+]i!RU07$FfnFhx7$$ \"3+++v=S2LKFfnFhx7$$\"3Jmmm\"p)=MLFfnFhx7$$\"3B++](=]@W$FfnFhx7$$\"35 L$e*[$z*RNFfnFhx7$$\"3e++]iC$pk$FfnFhx7$$\"3[m;H2qcZPFfnFhx7$$\"3O+]7. \"fF&QFfnFhx7$$\"3Ymm;/OgbRFfnFhx7$$\"3w**\\ilAFjSFfnFhx7$$\"3yLLL$)*p p;%FfnFhx7$$\"3)RL$3xe,tUFfnFhx7$$\"3Cn;HdO=yVFfnFhx7$$\"3a+++D>#[Z%Ff nFhx7$$\"3SnmT&G!e&e%FfnFhx7$$\"3#RLLL)Qk%o%FfnFhx7$$\"37+]iSjE!z%FfnF hx7$$\"3a+]P40O\"*[FfnFhx7$$\"\"&F)Fhx-%*AXESTICKSG6$%(DEFAULTG\"\"$-% *THICKNESSG6#Fgr-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$% \"tG%%f(t)G-%%VIEWG6$;F(F]]l;F($\"#D!\"\"" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 60 "There are two switching times for this piecewise functi on: " }{XPPEDIT 18 0 "t = 1;" "6#/%\"tG\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t = 3;" "6#/%\"tG\"\"$" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "u[1](t);" "6#-&%\"uG6#\"\"\"6#%\"tG" }{TEXT -1 12 " is used to " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 2*t;" "6#/-%\"fG6#%\"tG*&\"\"#\"\"\"F'F*" }{TEXT -1 8 " a nd to " }{TEXT 259 9 "switch on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) \+ = 2;" "6#/-%\"fG6#%\"tG\"\"#" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "u[3](t);" "6#-&%\"uG6#\"\"$6#%\"tG" }{TEXT -1 12 " is u sed to " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f( t) = 2*t;" "6#/-%\"fG6#%\"tG*&\"\"#\"\"\"F'F*" }{TEXT -1 8 " and to " }{TEXT 259 9 "switch on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1;" "6 #/-%\"fG6#%\"tG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus the function " }{XPPEDIT 18 0 "f (t);" "6#-%\"fG6#%\"tG" }{TEXT -1 50 " can be described using the unit step function as:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 2*t+u[1](t)*(-2*t+2)+u[3](t)*(-2+1);" "6#/-%\"fG6#%\"tG,(*&\" \"#\"\"\"F'F+F+*&-&%\"uG6#F+6#F'F+,&*&F*F+F'F+!\"\"F*F+F+F+*&-&F/6#\" \"$6#F'F+,&F*F4F+F+F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "2*t+u[1](t)*(2-2*t)-u[3](t);" "6#,(*&\"\"#\"\" \"%\"tGF&F&*&-&%\"uG6#F&6#F'F&,&F%F&*&F%F&F'F&!\"\"F&F&-&F+6#\"\"$6#F' F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Using the second shift formula we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t)] = L*[2*t+u[1](t)*(2-2*t)- u[3](t)];" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F%F&7#,(*&\"\"#F&F+F&F& *&-&%\"uG6#F&6#F+F&,&F0F&*&F0F&F+F&!\"\"F&F&-&F46#\"\"$6#F+F9F&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=L*[2*t]+L*[u[1](t)(2-2*t)]-L*[u[3]( t)] " "6#/%!G,(*&%\"LG\"\"\"7#*&\"\"#F(%\"tGF(F(F(*&F'F(7#--&%\"uG6#F( 6#F,6#,&F+F(*&F+F(F,F(!\"\"F(F(*&F'F(7#-&F26#\"\"$6#F,F(F8" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/s^2+exp( -s)*L*[2-2*(t+1)]-exp(-3*s)/s" "6#/%!G,(*&\"\"#\"\"\"*$%\"sGF'!\"\"F(* (-%$expG6#,$F*F+F(%\"LGF(7#,&F'F(*&F'F(,&%\"tGF(F(F(F(F+F(F(*&-F.6#,$* &\"\"$F(F*F(F+F(F*F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/s^2+exp(-s)*L*[-2*t]-exp(-3*s)/s" "6#/%!G,( *&\"\"#\"\"\"*$%\"sGF'!\"\"F(*(-%$expG6#,$F*F+F(%\"LGF(7#,$*&F'F(%\"tG F(F+F(F(*&-F.6#,$*&\"\"$F(F*F(F+F(F*F+F+" }{TEXT -1 3 " \n " } {XPPEDIT 18 0 "``=2/s^2-2*exp(-s)/s^2-exp(-3*s)/s" "6#/%!G,(*&\"\"#\" \"\"*$%\"sGF'!\"\"F(*(F'F(-%$expG6#,$F*F+F(*$F*F'F+F+*&-F.6#,$*&\"\"$F (F*F(F+F(F*F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(2-2*exp(-s))/s^2- exp(-3*s)/s" "6#/%!G,&*&,&\"\"#\"\"\"*&F(F)-%$expG6#,$%\"sG!\"\"F)F0F) *$F/F(F0F)*&-F,6#,$*&\"\"$F)F/F)F0F)F/F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 1 "u" }{TEXT -1 87 " from the subsection which follows the \+ examples and which implements the step function " }{XPPEDIT 18 0 "u[a] (t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 274 7 "u[a](t)" }{TEXT -1 46 " may be used to find the Laplace transform of " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 19 "collect_like_denoms" }{TEXT -1 85 " a lso from the subsection which follows the examples is used to simplify the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "f := t -> 2*t+'u[1](t)'*(2-2*t)-'u[3](t)':\n'f( t)'=f(t);\nconvert(rhs(%),Heaviside):\nf := unapply(%,t):\n'f(t)'=f(t) ;\n`Laplace transform`=inttrans[laplace](f(t),t,s);\n``=collect_like_d enoms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,(*&\" \"#\"\"\"F'F+F+*&-&%\"uG6#F+F&F+,&F*F+*&F*F+F'F+!\"\"F+F+-&F/6#\"\"$F& F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,**&\"\"#\"\"\"F'F +F+*&F*F+-%*HeavisideG6#,&F+!\"\"F'F+F+F+*(F*F+F-F+F'F+F1-F.6#,&\"\"$F 1F'F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\" \"#\"\"\"%\"sG!\"#F(*(F'F(-%$expG6#,$F)!\"\"F(F)F*F0*&-F-6#,$*&\"\"$F( F)F(F0F(F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&-%$expG6#,$* &\"\"$\"\"\"%\"sGF-!\"\"F-F.F/F/*(\"\"#F-,&F-F/-F(6#,$F.F/F-F-F.!\"#F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can sta rt with the piecewise definition of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6 #%\"tG" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIE CEWISE([2*t, t < 1],[2, 1 <= t and t < 3],[1, 3 <= t]);" "6#/-%\"fG6#% \"tG-%*PIECEWISEG6%7$*&\"\"#\"\"\"F'F.2F'F.7$F-31F.F'2F'\"\"$7$F.1F4F' " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 203 "f := t -> piecewise(t<1,2*t,t<3,2,t>=3,1):\n' f(t)'=f(t);\nsimplify(convert(f(t),Heaviside)):\nf := unapply(%,t):\n' f(t)'=f(t);\n`Laplace transform`=inttrans[laplace](f(t),t,s);\n``=coll ect_like_denoms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"tG-%*PIECEWISEG6%7$,$*&\"\"#\"\"\"F'F/F/2F'F/7$F.2F'\"\"$7$F/1F3F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,**&\"\"#\"\"\"F'F+F+* (F*F+-%*HeavisideG6#,&F'F+F+!\"\"F+F'F+F1*&F*F+F-F+F+-F.6#,&F'F+\"\"$F 1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\"\"# \"\"\"%\"sG!\"#F(*(F'F(-%$expG6#,$F)!\"\"F(F)F*F0*&-F-6#,$*&\"\"$F(F)F (F0F(F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"#\"\"\",&F( !\"\"-%$expG6#,$%\"sGF*F(F(F/!\"#F**&-F,6#,$*&\"\"$F(F/F(F*F(F/F*F*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplac e transform of the function " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG " }{TEXT -1 25 " defined by the formula: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([t, t < 1],[2 -t, 1 <= t and t < 3],[t-4, 3 <= t and t < 4],[0, 4 <= t]);" "6#/-%\"f G6#%\"tG-%*PIECEWISEG6&7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"31F-F'2F'\"\"$7$ ,&F'F-\"\"%F131F5F'2F'F87$\"\"!1F8F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f := t -> piecewise(t<1,t,t<3,2-t,t<4,t-4,0):\n'f(t)'=f(t);\nplot(f(t),t=0..6,y =-1.2..1.2,thickness=2,color=red,\n labels=[t,`f(t)` ],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*P IECEWISEG6&7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"2F'\"\"$7$,&\"\"%F1F'F-2F'F6 7$\"\"!%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 407 191 191 {PLOTDATA 2 "6(-%'CURVESG6#7Y7$$\"\"!F)F(7$$\"3%*******\\#HyI\"!#=F+7$ $\"33++]([kdW#F-F/7$$\"3++++v;\\DPF-F27$$\"3W+++D7$$\"3W++]( =x;N*F-FA7$$\"39++](Q=\"))**F-FD7$$\"35++D^=Di5!#<$\"3)*)***\\([\"[x$* F-7$$\"3(****\\P'=pD6FI$\"3M++]i83V()F-7$$\"33+++lN?c7FI$\"3B******\\V 'zV(F-7$$\"3-++]U$e6P\"FI$\"3%)*****\\d;%)G'F-7$$\"36+++&>q0]\"FI$\"3# *)*****\\!)H%*\\F-7$$\"3'******\\U80j\"FI$\"3Q+++]d'[p$F-7$$\"35+++0yt biUCF-7$$\"3)****\\(QNXp=FI$\"3B++]7YY08F-7$$\"3.+++X Dn/?FI$!3%z-+++XDn%!#?7$$\"3.+++!y?#>@FI$!3C++++y?#>\"F-7$$\"3'****\\( 3wY_AFI$!3h****\\(3wY_#F-7$$\"3#)******HOTqBFI$!3F)******HOTq$F-7$$\"3 7++v3\">)*\\#FI$!3I,+](3\">)*\\F-7$$\"3:++DEP/BEFI$!3_,+]isVIiF-7$$\"3 =++](o:;v#FI$!3&=++](o:;vF-7$$\"3=++v$)[opGFI$!3#>++v$)[op)F-7$$\"3&)* *\\7t;OLHFI$!3Z)**\\7t;OL*F-7$$\"3%*****\\i%Qq*HFI$!3W*****\\i%Qq**F-7 $$\"3%***\\i]2=jIFI$!3_++v$\\#>o$*F-7$$\"3&****\\(QIKHJFI$!3[++]7'pnq) F-7$$\"3#****\\7:xWC$FI$!3'3++v[G_b(F-7$$\"37++]Zn%)oLFI$!3t)****\\_K: J'F-7$$\"3y******4FL(\\$FI$!36-+++HnE]F-7$$\"3#)****\\d6.BOFI$!3y,++D% )opPF-7$$\"3(****\\(o3lWPFI$!3G++]78\\`DF-7$$\"3!*****\\A))ozQFI$!3)3+ +]x6J?\"F-7$$\"3(****\\iid.%RFI$!3G.++vtBkf!#>7$$\"3e******Hk-,SFIF(7$ $\"3S****\\FL!e1%FIF(7$$\"36+++D-eITFIF(7$$\"3u***\\(=_(zC%FIF(7$$\"3M +++b*=jP%FIF(7$$\"3g***\\(3/3(\\%FIF(7$$\"33++vB4JBYFIF(7$$\"3u*****\\ KCnu%FIF(7$$\"3s***\\(=n#f([FIF(7$$\"3P+++!)RO+]FIF(7$$\"30++]_!>w7&FI F(7$$\"3O++v)Q?QD&FIF(7$$\"3G+++5jyp`FIF(7$$\"3<++]Ujp-bFIF(7$$\"3++++ gEd@cFIF(7$$\"39++v3'>$[dFIF(7$$\"37++D6EjpeFIF(7$$\"\"'F)F(-%*AXESTIC KSG6$%(DEFAULTG\"\"$-%*THICKNESSG6#\"\"#-%'COLOURG6&%$RGBG$\"*++++\"! \")F(F(-%+AXESLABELSG6$%\"tG%%f(t)G-%%VIEWG6$;F(F^y;$!#7!\"\"$\"#7F\\[ l" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "There are three switching times for this piecewise function: " }{XPPEDIT 18 0 "t = 1;" "6#/%\" tG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "t=3" "6#/%\"tG\"\"$" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "t = 4;" "6#/%\"tG\"\"%" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "u[1](t);" "6#-&%\"uG6#\"\"\"6# %\"tG" }{TEXT -1 12 " is used to " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t;" "6#/-%\"fG6#%\"tGF'" }{TEXT -1 8 " and to " }{TEXT 259 9 "switch on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t ) = 2-t;" "6#/-%\"fG6#%\"tG,&\"\"#\"\"\"F'!\"\"" }{TEXT -1 2 ". " }} {PARA 15 "" 0 "" {XPPEDIT 18 0 "u[3](t);" "6#-&%\"uG6#\"\"$6#%\"tG" } {TEXT -1 12 " is used to " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 2-t;" "6#/-%\"fG6#%\"tG,&\"\"#\"\"\"F'!\"\"" } {TEXT -1 8 " and to " }{TEXT 259 9 "switch on" }{TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) = t-4;" "6#/-%\"fG6#%\"tG,&F'\"\"\"\"\"%!\"\"" } {TEXT -1 2 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "u[4](t);" "6#-&%\"uG 6#\"\"%6#%\"tG" }{TEXT -1 12 " is used to " }{TEXT 259 10 "switch off " }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t-4;" "6#/-%\"fG6#%\"tG,&F'\" \"\"\"\"%!\"\"" }{TEXT -1 8 " and to " }{TEXT 259 9 "switch on" } {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 0;" "6#/-%\"fG6#%\"tG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus the function " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\" tG" }{TEXT -1 50 " can be described using the unit step function as:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t+u[1](t)*(- t+2-t)+u[3](t)*(-(2-t)+``(t-4))+u[4](t)*(-(t-4));" "6#/-%\"fG6#%\"tG,* F'\"\"\"*&-&%\"uG6#F)6#F'F),(F'!\"\"\"\"#F)F'F1F)F)*&-&F-6#\"\"$6#F'F) ,&,&F2F)F'F1F1-%!G6#,&F'F)\"\"%F1F)F)F)*&-&F-6#F?6#F'F),$,&F'F)F?F1F1F )F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= t+u[1](t)*(2-2*t)+u[3](t)*(2*t-6)+u[4](t)*(4-t)" "6#/%!G,*%\"tG \"\"\"*&-&%\"uG6#F'6#F&F',&\"\"#F'*&F/F'F&F'!\"\"F'F'*&-&F+6#\"\"$6#F& F',&*&F/F'F&F'F'\"\"'F1F'F'*&-&F+6#\"\"%6#F&F',&F?F'F&F1F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Using the second shift formul a we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L*[f(t )] = L*[t+u[1](t)*(2-2*t)+u[3](t)*(2*t-6)+u[4](t)*(4-t)]" "6#/*&%\"LG \"\"\"7#-%\"fG6#%\"tGF&*&F%F&7#,*F+F&*&-&%\"uG6#F&6#F+F&,&\"\"#F&*&F6F &F+F&!\"\"F&F&*&-&F26#\"\"$6#F+F&,&*&F6F&F+F&F&\"\"'F8F&F&*&-&F26#\"\" %6#F+F&,&FFF&F+F8F&F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``= L*[t]+L*[u[ 1](t)*(2-2*t)]+L*[u[3](t)*(2*t-6)]+L*[u[4](t)*(4-t)]" "6#/%!G,**&%\"LG \"\"\"7#%\"tGF(F(*&F'F(7#*&-&%\"uG6#F(6#F*F(,&\"\"#F(*&F4F(F*F(!\"\"F( F(F(*&F'F(7#*&-&F06#\"\"$6#F*F(,&*&F4F(F*F(F(\"\"'F6F(F(F(*&F'F(7#*&-& F06#\"\"%6#F*F(,&FHF(F*F6F(F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/ s^2+exp(-s)*L*[2-2*(t+1)]+exp(-3*s)*L*[2*(t+3)-6]+exp(-4*s)*L*[4-(t+4) ]" "6#/%!G,**&\"\"\"F'*$%\"sG\"\"#!\"\"F'*(-%$expG6#,$F)F+F'%\"LGF'7#, &F*F'*&F*F',&%\"tGF'F'F'F'F+F'F'*(-F.6#,$*&\"\"$F'F)F'F+F'F1F'7#,&*&F* F',&F6F'F " 0 "" {MPLTEXT 1 0 217 "f := t -> t +'u[1](t)'*(2-2*t)+'u[3](t)'*(2*t-6)+'u[4](t)'*(4-t):\n'f(t)'=f(t);\nc onvert(rhs(%),Heaviside):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace \+ transform`=inttrans[laplace](f(t),t,s);\n``=collect_like_denoms(rhs(%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,*F'\"\"\"*&-&%\"u G6#F)F&F),&\"\"#F)*&F0F)F'F)!\"\"F)F)*&-&F-6#\"\"$F&F),&*&F0F)F'F)F)\" \"'F2F)F)*&-&F-6#\"\"%F&F),&F?F)F'F2F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,0F'\"\"\"*&\"\"#F)-%*HeavisideG6#,&F)!\"\"F'F)F)F )*(F+F)F,F)F'F)F0*(F+F)-F-6#,&\"\"$F0F'F)F)F'F)F)*&\"\"'F)F3F)F0*&\"\" %F)-F-6#,&F:F0F'F)F)F)*&F;F)F'F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%2Laplace~transformG,**&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*(F+F'-%$expG6# ,$F*F,F'F*!\"#F,*(F+F'-F/6#,$*&\"\"$F'F*F'F,F'F*F2F'*&-F/6#,$*&\"\"%F' F*F'F,F'F*F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,*\"\"\"!\" \"*&\"\"#F(-%$expG6#,$%\"sGF)F(F(*&F+F(-F-6#,$*&\"\"$F(F0F(F)F(F)-F-6# ,$*&\"\"%F(F0F(F)F(F(F0!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can start with the piecewise definition of " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 2 ": " }}{PARA 256 " " 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([t, t < 1],[2-t, 1 <= t and t < 3],[t-4, 3 <= t and t < 4],[0, 4 <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWI SEG6&7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"31F-F'2F'\"\"$7$,&F'F-\"\"%F131F5F '2F'F87$\"\"!1F8F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "f := t -> piecewise(t<1,t,t <3,2-t,t<4,t-4,t>=4,0):\n'f(t)'=f(t);\nconvert(f(t),Heaviside):\nf := \+ unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=inttrans[laplace](f(t ),t,s);\n``=collect_like_denoms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECEWISEG6&7$F'2F'\"\"\"7$,&\"\"#F-F'!\"\"2F' \"\"$7$,&\"\"%F1F'F-2F'F67$\"\"!1F6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,0F'\"\"\"*(\"\"#F)-%*HeavisideG6#,&F'F)F)!\"\"F)F'F) F0*&F+F)F,F)F)*&\"\"'F)-F-6#,&F'F)\"\"$F0F)F0*(F+F)F4F)F'F)F)*&\"\"%F) -F-6#,&F:F0F'F)F)F)*&F;F)F'F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2 Laplace~transformG,**&\"\"\"F'*$)%\"sG\"\"#F'!\"\"F'*(F+F'-%$expG6#,$F *F,F'F*!\"#F,*(F+F'-F/6#,$*&\"\"$F'F*F'F,F'F*F2F'*&-F/6#,$*&\"\"%F'F*F 'F,F'F*F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,*\"\"\"!\"\"*& \"\"#F(-%$expG6#,$%\"sGF)F(F(*&F+F(-F-6#,$*&\"\"$F(F0F(F)F(F)-F-6#,$*& \"\"%F(F0F(F)F(F(F0!\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We find the Laplace transform of the function " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 25 " defined by the \+ formula: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([1, t < 2*Pi],[1+sin*t, 2*Pi <= t and \+ t < 4*Pi],[1, 4*Pi <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$\"\"\"2F '*&\"\"#F,%#PiGF,7$,&F,F,*&%$sinGF,F'F,F,31*&F/F,F0F,F'2F'*&\"\"%F,F0F ,7$F,1*&F:F,F0F,F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "f := t -> piecewise(t<2*Pi,1,t<4*Pi,1+sin(t),t>=4*Pi,1);\nplot(f( t),t=0..6*Pi,thickness=2,color=COLOR(RGB,0.9,0.6,0),\n \+ labels=[t,`f(t)`],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6(29$,$*&\"\"# \"\"\"%#PiGF4F4F42F0,$*&\"\"%F4F5F4F4,&F4F4-%$sinG6#F0F41F7F0F4F(F(F( " }}{PARA 13 "" 1 "" {GLPLOT2D 358 192 192 {PLOTDATA 2 "6(-%'CURVESG6# 7jp7$$\"\"!F)$\"\"\"F)7$$\"3vqcubnm3T!#=F*7$$\"39;#)*4t&f$o(F/F*7$$\"3 IX>QDxRq6!#F6F*7$$\"3o\"=B4x 38N#F6F*7$$\"3sgfzH@(zt#F6F*7$$\"3KS)p'G*fy8$F6F*7$$\"3h%>,BGlk`$F6F*7 $$\"3_%)pL$))zk%RF6F*7$$\"3;3\\5n4i2VF6F*7$$\"3wOXR\"H!=9ZF6F*7$$\"3s9 6L()*3C7&F6F*7$$\"34/R!)yH\"e^&F6F*7$$\"3#R^?`BPWp&F6F*7$$\"3!H7P=\\hI (eF6F*7$$\"3aR-z]AEzfF6F*7$$\"3=cLu4IY&3'F6F*7$$\"3^9*>#*Qj&QhF6F*7$$ \"3#GZ'poPm\">'F6F*7$$\"3W_ZVeR@=iF6F*7$$\"3;JI<[TwWiF6F*7$$\"3_q@/V#R !eiF6F*7$$\"3))48\"zL98F'F6F*7$$\"37]/yK%*e%G'F6$\"3Tv%R^7/9+\"F67$$\" 3[*e\\w_kyH'F6$\"3+NrH$pyY,\"F67$$\"3%49pSZ'zxkF6$\"3'=H2l0&Q$>\"F67$$ \"3_\"p)[?%Gxl'F6$\"3[\\PKat%eO\"F67$$\"3yA$G#\\?.noF6$\"3!yhP-`P7b\"F 67$$\"3/az'znNj2(F6$\"3mPZSF67$$\"3A#3[i&)3&[vF6$\"3uNq/u[q`>F67$$ \"3%e&*)Q\"oV,l(F6$\"3UE'=\"yoHz>F67$$\"3m#RfR4h4q(F6$\"3y'QgT=:$))>F6 7$$\"3[H)Hl]yF67$$\"3Hm-5>ff-yF6$\"35ZgjW+o)*>F6 7$$\"35.2nJLT`yF6$\"33]i^Q)*****>F67$$\"3Y/!4d_.=!zF6$\"3%)*=Q7vc))*>F 67$$\"3%\\IZ(>P>]zF6$\"3qc_\\'=v`*>F67$$\"3S0cy8Re)*zF6$\"37Fm;&Hj&*)> F67$$\"3w1R#y5up/)F6$\"35`[y$oM9)>F67$$\"3[40!f\\aP9)F6$\"35!3J9!)3$e> F67$$\"3?7r(R)[`S#)F6$\"3'[c(oxR@E>F67$$\"3>aT.[_\\U%)F6$\"3yt93GnwJ=F 67$$\"3<'>\"47cXW')F6$\"3<3tpW&3Nq\"F67$$\"3O\"*z\\r#=*H))F6$\"3aeC'3b #Rg:F67$$\"3w%y/4$4Q:!*F6$\"3KE%*3fi0)R\"F67$$\"3C;+@kuU:#*F6$\"3W0nZ@ Y#y?\"F67$$\"3sZ_^(*RZ:%*F6$\"3/\\P-HQI45F67$$\"3SY.ag+P>&*F6$\"3s(G'3 -!*[b!*F/7$$\"3'oWlN7mKi*F6$\"3>\\h*z'f7G!)F/7$$\"3KZ0f'=irs*F6$\"3yL6 =v*G?-(F/7$$\"3yZch\\#e5$)*F6$\"3)zN)Gj%[![gF/7$$\"3hy%=1>%>,5!#;$\"3c 2;7X!y*fWF/7$$\"3W#Rvib#G>5Fiy$\"3+uqKYtn_IF/7$$\"3iH=t'[=)Q5Fiy$\"3gJ 2h1Ef)y\"F/7$$\"3!oE)=7$$\"3#>5l5oX%o5Fiy$\" 3V^y@k?z+[F[[l7$$\"3/P>%\\%p`y5Fiy$\"3/\"4;Lj!=,AF[[l7$$\"3_a.)od#e$3 \"Fiy$\"3EJtAI#oKF\"F[[l7$$\"3)>x=)3#G')3\"Fiy$\"3?;l#p()\\k'f!#?7$$\" 3Y*=d2%Qn$4\"Fiy$\"3k%G5#\\6LI()4\"Fiy$\"3C7S!fDJ3^ $!#A7$$\"3-rh/*['R)!#@7$$\"37Nnl0=f36Fiy$\"3as(ogHB# ySF`\\l7$$\"3A*HP@(z_86Fiy$\"3D&[,V1UHu*F`\\l7$$\"39jyhQTY=6Fiy$\"3YPN ]]0+#y\"F[[l7$$\"3:\"**y:ZO$G6Fiy$\"3A[Zx?Km7TF[[l7$$\"3<>,a/)3#Q6Fiy$ \"3qjJ!eX8rP(F[[l7$$\"3'zqrSy7t:\"Fiy$\"3do!e9V\"*>i\"F/7$$\"3s'H.Ow;k <\"Fiy$\"3;h2m7#36\"GF/7$$\"3=\"yLdZGw>\"Fiy$\"3DhDYqjnNWF/7$$\"3ilU'y =S)=7Fiy$\"3P$y/\"ylm4jF/7$$\"3L:Dka)**yB\"Fiy$\"3c\")\\.YRAP\")F/7$$ \"3/l2U@&fpD\"FiyF*7$$\"3oj4\"Ry4tF\"FiyF*7$$\"3Ki6SY+m(H\"FiyF*7$$\"3 xS3Kr2aM8FiyF*7$$\"3+cS8X6'[P\"FiyF*7$$\"3%G3_YZ*z79FiyF*7$$\"3c#=J/'f X_9FiyF*7$$\"3s;f#QUF7\\\"FiyF*7$$\"3/))z%Gb<=`\"FiyF*7$$\"3]5vMr1\"4d \"FiyF*7$$\"3Q&[d8.*)3h\"FiyF*7$$\"3CR>O]j`];FiyF*7$$\"3epd\"*=\"opo\" FiyF*7$$\"3NJ9+PIsG_F<'y)e!=FiyF* 7$$\"3'GJ&Rl%**R%=FiyF*7$$\"3!*****Q)eb\\)=FiyF*-%*AXESTICKSG6$%(DEFAU LTG\"\"$-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$%\"tG%%f(t)G-%&COLORG6&%$R GBG$\"\"*!\"\"$\"\"'F^elF)-%%VIEWG6$;F($\"+#fb\\)=!\")F]dl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "There are two switching times for this \+ piecewise function: " }{XPPEDIT 18 0 "t = 2*Pi;" "6#/%\"tG*&\"\"#\"\" \"%#PiGF'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "t = 4*Pi;" "6#/%\"tG* &\"\"%\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "u[2*Pi](t);" "6#-&%\"uG6#*&\"\"#\"\"\"%#PiGF)6#%\"tG" }{TEXT -1 12 " \+ is used to " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1;" "6#/-%\"fG6#%\"tG\"\"\"" }{TEXT -1 8 " and to " }{TEXT 259 9 "switch on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1+sin*t;" "6# /-%\"fG6#%\"tG,&\"\"\"F)*&%$sinGF)F'F)F)" }{TEXT -1 1 "." }}{PARA 15 " " 0 "" {XPPEDIT 18 0 "u[4*Pi](t);" "6#-&%\"uG6#*&\"\"%\"\"\"%#PiGF)6#% \"tG" }{TEXT -1 12 " is used to " }{TEXT 259 10 "switch off" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1+sin*t;" "6#/-%\"fG6#%\"tG,&\"\"\"F)* &%$sinGF)F'F)F)" }{TEXT -1 8 " and to " }{TEXT 259 9 "switch on" } {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1;" "6#/-%\"fG6#%\"tG\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus the function " }{XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\" tG" }{TEXT -1 50 " can be described using the unit step function as:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = 1+u[2*Pi](t) *(-1+1+sin*t)+u[4*Pi](t)*(-(1+sin*t)+1);" "6#/-%\"fG6#%\"tG,(\"\"\"F)* &-&%\"uG6#*&\"\"#F)%#PiGF)6#F'F),(F)!\"\"F)F)*&%$sinGF)F'F)F)F)F)*&-&F -6#*&\"\"%F)F1F)6#F'F),&,&F)F)*&F6F)F'F)F)F4F)F)F)F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1+u[2*Pi](t)*sin* t-u[4*Pi](t)*sin*t;" "6#,(\"\"\"F$*(-&%\"uG6#*&\"\"#F$%#PiGF$6#%\"tGF$ %$sinGF$F.F$F$*(-&F(6#*&\"\"%F$F,F$6#F.F$F/F$F.F$!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Using the second shift formula we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "L*[f(t)] = L*[1+u[2*pi](t)*sin*t-u[4*Pi](t)*sin*t]; " "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&F%F&7#,(F&F&*(-&%\"uG6#*&\"\"#F& %#piGF&6#F+F&%$sinGF&F+F&F&*(-&F26#*&\"\"%F&%#PiGF&6#F+F&F8F&F+F&!\"\" F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = L*[1]+L*[u[2*Pi](t)*sin*t]-L*[u[4*Pi](t)*sin*t];" "6#/%!G,(*&%\" LG\"\"\"7#F(F(F(*&F'F(7#*(-&%\"uG6#*&\"\"#F(%#PiGF(6#%\"tGF(%$sinGF(F5 F(F(F(*&F'F(7#*(-&F/6#*&\"\"%F(F3F(6#F5F(F6F(F5F(F(!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/s+exp(-2*P i*s)*L*[sin(t+2*Pi)]-exp(-4*Pi*s)*L*[sin(t+4*Pi)]" "6#/%!G,(*&\"\"\"F' %\"sG!\"\"F'*(-%$expG6#,$*(\"\"#F'%#PiGF'F(F'F)F'%\"LGF'7#-%$sinG6#,&% \"tGF'*&F0F'F1F'F'F'F'*(-F,6#,$*(\"\"%F'F1F'F(F'F)F'F2F'7#-F56#,&F8F'* &F?F'F1F'F'F'F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/s+exp(-2*Pi*s)*L*[sin*t]-exp(-4*Pi*s)*L*[sin*t]; " "6#/%!G,(*&\"\"\"F'%\"sG!\"\"F'*(-%$expG6#,$*(\"\"#F'%#PiGF'F(F'F)F' %\"LGF'7#*&%$sinGF'%\"tGF'F'F'*(-F,6#,$*(\"\"%F'F1F'F(F'F)F'F2F'7#*&F5 F'F6F'F'F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/s+exp(-2*Pi*s)/(s^2+1)-exp(-4*Pi*s)/(s^2+1)" "6#/% !G,(*&\"\"\"F'%\"sG!\"\"F'*&-%$expG6#,$*(\"\"#F'%#PiGF'F(F'F)F',&*$F(F 0F'F'F'F)F'*&-F,6#,$*(\"\"%F'F1F'F(F'F)F',&*$F(F0F'F'F'F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/s+(exp(-2*Pi*s)-exp(-4*Pi*s))/(s^2+1);" " 6#/%!G,&*&\"\"\"F'%\"sG!\"\"F'*&,&-%$expG6#,$*(\"\"#F'%#PiGF'F(F'F)F'- F-6#,$*(\"\"%F'F2F'F(F'F)F)F',&*$F(F1F'F'F'F)F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 1 "u" }{TEXT -1 87 " from the subsection which follows the examples and which implements the step function " }{XPPEDIT 18 0 "u[a](t)=`` " "6#/-&%\"uG6#%\"aG6#%\"tG%!G" }{TEXT 274 7 "u[a](t)" } {TEXT -1 46 " may be used to find the Laplace transform of " } {XPPEDIT 18 0 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 19 "collect_like_denoms" } {TEXT -1 85 " also from the subsection which follows the examples is u sed to simplify the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "f := t -> 1+'u[2*Pi](t)'*si n(t)-'u[4*Pi](t)'*sin(t):\n'f(t)'=f(t);\nsimplify(convert(rhs(%),Heavi side)):\nf := unapply(%,t):\n'f(t)'=f(t);\n`Laplace transform`=inttran s[laplace](f(t),t,s);\n``=collect_like_denoms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,(\"\"\"F)*&-&%\"uG6#,$*&\"\"#F)%#Pi GF)F)F&F)-%$sinGF&F)F)*&-&F-6#,$*&\"\"%F)F2F)F)F&F)F3F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,(\"\"\"F)*&-%*HeavisideG6#,&* &\"\"#F)%#PiGF)!\"\"F'F)F)-%$sinGF&F)F)*&-F,6#,&*&\"\"%F)F1F)F2F'F)F)F 3F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Laplace~transformG,(*&\"\" \"F'%\"sG!\"\"F'*&-%$expG6#,$*(\"\"#F'F(F'%#PiGF'F)F',&*$)F(F0F'F'F'F' F)F'*&-F,6#,$*(\"\"%F'F(F'F1F'F)F'F2F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&,&-%$expG6#,$*(\"\"#\"\"\"%\"sGF.%#PiGF.!\"\"F.-F)6#,$*( \"\"%F.F/F.F0F.F1F1F.,&*$)F/F-F.F.F.F.F1F.*&F.F.F/F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can start with the pi ecewise definition of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 2 ": " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(t) = PIECEWISE([1 , t < 2*Pi],[1+sin*t, 2*Pi <= t and t < 4*Pi],[1, 4*Pi <= t]);" "6#/-% \"fG6#%\"tG-%*PIECEWISEG6%7$\"\"\"2F'*&\"\"#F,%#PiGF,7$,&F,F,*&%$sinGF ,F'F,F,31*&F/F,F0F,F'2F'*&\"\"%F,F0F,7$F,1*&F:F,F0F,F'" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "f := t -> piecewise(t<2*Pi,1,t<4*Pi,1+sin(t),t>=4*Pi,1):\n'f( t)'=f(t);\nconvert(f(t),Heaviside):\nf := unapply(%,t):\n'f(t)'=f(t); \n`Laplace transform`=inttrans[laplace](f(t),t,s);\n``=collect_like_de noms(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIEC EWISEG6%7$\"\"\"2F',$*&\"\"#F,%#PiGF,F,7$,&-%$sinGF&F,F,F,2F',$*&\"\"% F,F1F,F,7$F,1F7F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG,( \"\"\"F)*&-%*HeavisideG6#,&F'F)*&\"\"#F)%#PiGF)!\"\"F)-%$sinGF&F)F)*&- F,6#,&F'F)*&\"\"%F)F1F)F2F)F3F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %2Laplace~transformG,(*&\"\"\"F'%\"sG!\"\"F'*&-%$expG6#,$*(\"\"#F'F(F' %#PiGF'F)F',&*$)F(F0F'F'F'F'F)F'*&-F,6#,$*(\"\"%F'F(F'F1F'F)F'F2F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&,&-%$expG6#,$*(\"\"#\"\"\"% \"sGF.%#PiGF.!\"\"F.-F)6#,$*(\"\"%F.F/F.F0F.F1F1F.,&*$)F/F-F.F.F.F.F1F .*&F.F.F/F1F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "A pro cedure " }{TEXT 0 1 "u" }{TEXT -1 41 " which implements the unit step \+ function " }{XPPEDIT 18 0 "u[a](t)" "6#-&%\"uG6#%\"aG6#%\"tG" }{TEXT -1 75 " and a procedure for \ncollecting together terms with the same \+ denominator: " }{TEXT 0 19 "collect_like_denoms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "u := proc(t )\n local a;\n if type(procname,specindex(algebraic,u)) and type(t ,algebraic) then\n a := op(1,procname); \n piecewise(t=a,1);\n else 'procname'(t)\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 600 "collec t_like_denoms := proc(ee::algebraic)\n local i,j,denoms,terms,term,d n;\n if op(0,ee)=`+` then\n denoms := \{\};\n for i to nop s(ee) do\n denoms := \{op(denoms),denom(op(i,ee))\};\n en d do;\n end if;\n denoms := [op(denoms)];\n terms := [];\n for i to nops(denoms) do\n dn := op(i,denoms);\n term := 0;\n \+ for j to nops(ee) do\n if denom(op(j,ee))=dn then\n \+ term := term+op(j,ee);\n end if;\n end do;\n te rms := [op(terms),term];\n end do;\n terms := map(normal,terms); \n add(op(i,terms),i=1..nops(terms)); \nend proc:" }}} {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 8 "Summary " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Each stage of the form " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = PIECEWISE([`.`, ``],[g(t) , ` . . . ` and t < a],[h(t), a <= t and ` . . . `],[`.`, ``]);" "6#/- %\"fG6#%\"tG-%*PIECEWISEG6&7$%\".G%!G7$-%\"gG6#F'3%(~.~.~.~G2F'%\"aG7$ -%\"hG6#F'31F5F'F37$F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "in the formula for a piecewise function " }{XPPEDIT 18 0 "f(t)" "6 #-%\"fG6#%\"tG" }{TEXT -1 17 " leads to a term " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = ` . . . `+u[a](t)*(-g(t)+h(t))+` . . . `" "6#/-%\"fG6#%\"tG,(%(~.~.~.~G\"\"\"*&-&%\"uG6#%\"aG6#F'F*,&- %\"gG6#F'!\"\"-%\"hG6#F'F*F*F*F)F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 40 " | | " }}{PARA 256 " " 0 "" {TEXT -1 24 " \"" }{TEXT 271 10 "switch o ff" }{TEXT -1 9 "\" \"" }{TEXT 272 9 "switch on" }{TEXT -1 2 "\" " }}{PARA 0 "" 0 "" {TEXT -1 22 "in the description of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 26 " using the step function. \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 341 186 186 {PLOTDATA 2 "60-%'CURVESG6$7S7$$!\"\"\"\"!$\"3++++++++v!#=7$$!3!RLL$e%G?y*F-$\"3 8>*35)zz2wF-7$$!3OmmT&esBf*F-$\"3UpOc/(f'*p(F-7$$!3yKL$3s%3z$*F-$\"3Ho d)\\C>3!yF-7$$!33LL$e/$Qk\"*F-$\"3wj5t%3_.!zF-7$$!3!pm;/\"=q]*)F-$\"3i %4DvUBr*zF-7$$!3'GL$3_>f_()F-$\"3-_\"3IN.[3)F-7$$!3))***\\(o1YZ&)F-$\" 3]&H/H!H_t\")F-7$$!30ML3-OJN$)F-$\"3y8J()yO1j#)F-7$$!3p***\\P*o%Q7)F-$ \"3)>rs6zx+N)F-7$$!3ammm\"RFj!zF-$\"3'R`%Rz'\\sV)F-7$$!3JLL$e4OZr(F-$ \"3!f:\"Hu627&)F-7$$!3=+++v'\\!*\\(F-$\"3GfqH\\j5%f)F-7$$!33+++DwZ#G(F -$\"3ceV.\"*z8u')F-7$$!3[*****\\KqP2(F-$\"3+N(GZLW!\\()F-7$$!3OLL3-TC% )oF-$\"3<_ F-$\"3g^+HFF_>$*F-7$$!3VLL$e*e$\\+&F-$\"3Au\\'pTlPP*F-7$$!3;ML3-;Y%y%F -$\"3ZaZXzJsF%*F-7$$!3]++D\"3QDf%F-$\"3q8nI\\[rs%*F-7$$!3'QLL3Ub_Q%F-$ \"3L'H^BPQ#>&*F-7$$!3O+++]@6rTF-$\"3?fI!eeX]c*F-7$$!3/,+]PZhhRF-$\"3Xp 0z;-k2'*F-7$$!3y++v=_\"*ePF-$\"3uSiX4RwY'*F-7$$!3*3++D'>&Q`$F-$\"3?s$y wD(z(o*F-7$$!3Pnmm;EiJLF-$\"30zG]os]A(*F-7$$!3?+++D'*p:JF-$\"3!>PphR5t v*F-7$$!3zLL3-8/?HF-$\"3+(=`)pR$oy*F-7$$!3<+++v]81FF-$\"3+pj*Q#3#p\")* F-7$$!3snmT&)f'[]#F-$\"3Akv()f69V)*F-7$$!3g++v$z\"[%H#F-$\"3IWPXKQQo)* F-7$$!33nmm\"z#z)3#F-$\"3-Mq$ohB4*)*F-7$$!3Y++voaXt=F-$\"3nn_;:TD7**F- 7$$!33LLLL+1m;F-$\"3XLK8*411$**F-7$$!3iLL$eCoRX\"F-$\"3-4F]3%\\r%**F-7 $$!3$om;aoKOC\"F-$\"3;$3%fVWLh**F-7$$!3E+++]hN]5F-$\"3CHR&*)z=C(**F-7$ $!3anmm\"H%R)G)!#>$\"3Smk,Ic#G)**F-7$$!30PLLLB72jFby$\"3'>4o>0b+***F-7 $$!3i(***\\(=tY>%Fby$\"3m) )***F-7$$F*F*$\"\"\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")FezFez-F$6$7S7$F ez$\"3++++++++:!#<7$$\"3emmm;arz@Fby$\"3sev,@7))*\\\"Fe[l7$$\"3[LL$e9u i2%Fby$\"3Q+ss*f%e*\\\"Fe[l7$$\"3nmmm\"z_\"4iFby$\"3OV-agh.*\\\"Fe[l7$ $\"3[mmmT&phN)Fby$\"39tZwgVD)\\\"Fe[l7$$\"3CLLe*=)H\\5F-$\"3wULFLuC(\\ \"Fe[l7$$\"3gmm\"z/3uC\"F-$\"3%=)\\!H$*4h\\\"Fe[l7$$\"3%)***\\7LRDX\"F -$\"3WHzsB`s%\\\"Fe[l7$$\"3]mm\"zR'ok;F-$\"33y9*z/sI\\\"Fe[l7$$\"3w*** \\i5`h(=F-$\"3=rZ!Q7+7\\\"Fe[l7$$\"3WLLL3En$4#F-$\"3o'yAvLT!*[\"Fe[l7$ $\"3qmm;/RE&G#F-$\"3K#y?A#R%p[\"Fe[l7$$\"3\")*****\\K]4]#F-$\"3%fqz'=J O%[\"Fe[l7$$\"3$******\\PAvr#F-$\"3#fV`.oP:[\"Fe[l7$$\"3)******\\nHi#H F-$\"3etGs\\Hfy9Fe[l7$$\"3jmm\"z*ev:JF-$\"3%Q\\hH;IdZ\"Fe[l7$$\"3?LLL3 47TLF-$\"3sW(*oxA4s9Fe[l7$$\"3,LLLLY.KNF-$\"3gMBPG=\")o9Fe[l7$$\"3w*** \\7o7Tv$F-$\"3y%>T\\fmZY\"Fe[l7$$\"3'GLLLQ*o]RF-$\"3GV5\\L,)4Y\"Fe[l7$ $\"3A++D\"=lj;%F-$\"3-p6%H].mX\"Fe[l7$$\"31++vV&RAX\"F e[l7$$\"3WLL$e9Ege%F-$\"3uKVZ54UZ9Fe[l7$$\"3GLLeR\"3Gy%F-$\"3bQ)\\d'= \"GW\"Fe[l7$$\"3cmm;/T1&*\\F-$\"34k\")[OLiP9Fe[l7$$\"3&em;zRQb@&F-$\"3 /U'\\!)R&*>V\"Fe[l7$$\"3\\***\\(=>Y2aF-$\"3XrJ**)Q)*oU\"Fe[l7$$\"39mm; zXu9cF-$\"3K'zw#3m=@9Fe[l7$$\"3l******\\y))GeF-$\"3%fI!3m,1:9Fe[l7$$\" 3'*)***\\i_QQgF-$\"36dSbeZ%)39Fe[l7$$\"3@***\\7y%3TiF-$\"36CJ)=:AES\"F e[l7$$\"35****\\P![hY'F-$\"3EPG*QKsaR\"Fe[l7$$\"3kKLL$Qx$omF-$\"3;@Ood =$))Q\"Fe[l7$$\"3!)*****\\P+V)oF-$\"3EPp'3-;:Q\"Fe[l7$$\"3?mm\"zpe*zqF -$\"3G&[*3iaou8Fe[l7$$\"3%)*****\\#\\'QH(F-$\"3!ojRh$))*pO\"Fe[l7$$\"3 GKLe9S8&\\(F-$\"3')*ee_Td&f8Fe[l7$$\"3R***\\i?=bq(F-$\"3cuG$HZi:N\"Fe[ l7$$\"3\"HLL$3s?6zF-$\"3'o.n7+KNM\"Fe[l7$$\"3a***\\7`Wl7)F-$\"3!p-a\\= )*[L\"Fe[l7$$\"3#pmmm'*RRL)F-$\"3%**)*z:hjjK\"Fe[l7$$\"3Qmm;a<.Y&)F-$ \"3kP>9`LT<8Fe[l7$$\"3=LLe9tOc()F-$\"3xT-jy]J38Fe[l7$$\"3u******\\Qk\\ *)F-$\"3%HR&R(of(*H\"Fe[l7$$\"3CLL$3dg6<*F-$\"3#*z*fW`C(*G\"Fe[l7$$\"3 ImmmmxGp$*F-$\"3%fZjo6T0G\"Fe[l7$$\"3A++D\"oK0e*F-$\"3k^(eQ[L0F\"Fe[l7 $$\"3A++v=5s#y*F-$\"3WnKnBfug7Fe[l7$Ffz$\"3+++++++]7Fe[lFhz-F$6%7$7$F( Fez7$FfzFez-%*LINESTYLEG6#Fgz-Fiz6&F[[lF*F*F*-F$6%7$7$FezFez7$Fez$\"33 +++++++;Fe[l-Fjjl6#\"\"#F\\[m-F$6&7#FdzFhz-%'SYMBOLG6#%'CIRCLEG-%&STYL EG6#%&POINTG-F$6&7#Fb[lF[\\mFhzF_\\m-F$6&Fe\\m-F\\\\m6#%(DIAMONDGFhzF_ \\m-F$6&Fe\\m-F\\\\m6#%&CROSSGFhzF_\\m-%%TEXTG6%7$Fez$!\"&!\"#Q\"a6\"F \\[m-Fa]m6%7$$!#bFf]m$\"$1\"Ff]mQ%g(t)Fh]mFhz-Fa]m6%7$$\"#bFf]m$\"$c\" Ff]mQ%h(t)Fh]mFhz-%+AXESLABELSG6%Q\"tFh]mQ!Fh]m-%%FONTG6#%(DEFAULTG-%* AXESSTYLEG6#%%NONEG-%%VIEWG6$;F(FfzFa_m" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Inverse version of the second shift formula: \nif " }{XPPEDIT 18 0 "L*[f(t)] = F(s);" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&-%\"FG6#%\"sG" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "L^(-1)*[exp(-a*s)*F(s)] = u[a]( t)*f(t-a);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&-%$expG6#,$*&%\"aGF(%\"sGF( F)F(-%\"FG6#F2F(F(*&-&%\"uG6#F16#%\"tGF(-%\"fG6#,&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 266 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "L^(-1)*[exp(-2*s)/(s^3)];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&-%$expG 6#,$*&\"\"#F'%\"sGF'F(F'*$F1\"\"$F(F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 8 "Solution" }{TEXT 267 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "First find " }{XPPEDIT 18 0 "f( t) = L^(-1)*[1/(s^3)];" "6#/-%\"fG6#%\"tG*&)%\"LG,$\"\"\"!\"\"F,7#*&F, F,*$%\"sG\"\"$F-F," }{XPPEDIT 18 0 "`` = t^2/2;" "6#/%!G*&%\"tG\"\"#F' !\"\"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[exp(-2*s)/(s^3)] = \+ u[2](t)*f(t-2);" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&-%$expG6#,$*&\"\"#F(% \"sGF(F)F(*$F2\"\"$F)F(*&-&%\"uG6#F16#%\"tGF(-%\"fG6#,&F;F(F1F)F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = u[2](t);" "6#/%!G-&%\"uG6#\"\"#6#% \"tG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(t-2)^2/2" "6#*&,&%\"tG\"\"\"\" \"#!\"\"F'F'F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=PIECEWISE([0 , t< 2],[(t-2)^2/2 , t>=2])" "6#/%!G-%*PIECEWISEG6$7$\"\"!2%\"tG\"\"#7$*&,& F+\"\"\"F,!\"\"F,F,F11F,F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 0 10 "invlaplace " }{TEXT -1 11 " we obtain:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "exp(-2*s)/s^3;\n`inverse transform `=inttrans[invlaplace](%,s,t);\n'f(t)'=convert(rhs(%),piecewise);\nf : = unapply(rhs(%),t):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#, $*&\"\"#\"\"\"%\"sGF*!\"\"F*F+!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%2inverse~transformG,$*&#\"\"\"\"\"#F(*&-%*HeavisideG6#,&%\"tGF(F)!\" \"F()F.F)F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PI ECEWISEG6%7$\"\"!2F'\"\"#7$%*undefinedG/F'F.7$,$*&F.!\"\",&F'\"\"\"F.F 5F.F72F.F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The graph can easily be plotted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(f(t),t=0..4,thickn ess=2,labels=[t,`f(t)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 347 171 171 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>F(7$$\"3R LLLe'40j\"!#=F(7$$\"3mmmm;6m$[#F1F(7$$\"3fmmm;yYULF1F(7$$\"3%HLL$eF>(> %F1F(7$$\"3Qmmm\">K'*)\\F1F(7$$\"3P*****\\Kd,\"eF1F(7$$\"3-mmm\"fX(emF 1F(7$$\"3.*****\\U7Y](F1F(7$$\"3'QLLLV!pu$)F1F(7$$\"3xmmm;c0T\"*F1F(7$ $\"3#*******H,Q+5!#FPF(7$$\"3immmTc-)*>FPF(7$$ \"3Mmm;f`@'3#FP$\"3.L)=\"ySa;P!#?7$$\"3y****\\nZ)H;#FP$\"3\"3W^=s,#G8F -7$$\"3YmmmJy*eC#FP$\"3U@%=4=(GBIF-7$$\"3')******R^bJBFP$\"3;')4-V0W' \\&F-7$$\"3f*****\\5a`T#FP$\"319v,F;&fi)F-7$$\"3o****\\7RV'\\#FP$\"3n& G+u9LAB\"F17$$\"3k*****\\@fke#FP$\"3[y?Ha?n>$GFP$\"3' \\NJ![D)4Y$F17$$\"3$*******pfa;`.`%4UF17$$\"3#HLLeg`!)*HFP$ \"3c&e%e+]b!)\\F17$$\"3w****\\#G2A3$FP$\"33p,[6I'e&eF17$$\"3;LLL$)G[kJ FP$\"3_l:)y#>5!y'F17$$\"3#)****\\7yh]KFP$\"38A9rkXA?yF17$$\"3xmmm')fdL LFP$\"3;l)p5cC@*))F17$$\"3bmmm,FT=MFP$\"30mY7'HZf+\"FP7$$\"3FLL$e#pa-N FP$\"3X+c;KO#)G6FP7$$\"3!*******Rv&)zNFP$\"3KDuMB\\(zC\"FP7$$\"3ILLLGU YoOFP$\"3OsRhSk)=R\"FP7$$\"3_mmm1^rZPFP$\"3xQa.ZSDF:FP7$$\"34++]sI@KQF P$\"3SX*>:P-&y;FP7$$\"34++]2%)38RFP$\"3)GWbui`*H=FP7$$\"\"%F)$\"\"#F)- %'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#Fgw-%+AXESLABELSG6$%\"tG %%f(t)G-%%VIEWG6$;F(Fdw%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "L^(-1)*[e xp(-Pi*s/4)/(s^2+4)];" "6#*&)%\"LG,$\"\"\"!\"\"F'7#*&-%$expG6#,$*(%#Pi GF'%\"sGF'\"\"%F(F(F',&*$F1\"\"#F'F2F'F(F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT 264 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "First find " }{XPPEDIT 18 0 "f(t) = L^(-1)*[1/(s^2+4)];" "6#/-%\"fG6#%\"tG*&)%\"LG,$\"\"\"!\"\"F ,7#*&F,F,,&*$%\"sG\"\"#F,\"\"%F,F-F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sin*2*t/2;" "6#**%$sinG\"\"\"\"\"#F%%\"tGF%F&!\"\"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L^(-1)*[exp(-Pi*s/4)/(s^2+4)] = u[Pi/4](t)*f(t-3 );" "6#/*&)%\"LG,$\"\"\"!\"\"F(7#*&-%$expG6#,$*(%#PiGF(%\"sGF(\"\"%F)F )F(,&*$F2\"\"#F(F3F(F)F(*&-&%\"uG6#*&F1F(F3F)6#%\"tGF(-%\"fG6#,&F>F(\" \"$F)F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "u[Pi/4](t)*sin(2*(t-Pi/4));" "6#*&-&%\"uG6#*&%#PiG\"\" \"\"\"%!\"\"6#%\"tGF*-%$sinG6#*&\"\"#F*,&F.F**&F)F*F+F,F,F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u[Pi/4](t)*sin(2*t-Pi/2);" "6#*&-&%\"uG6#*&%# PiG\"\"\"\"\"%!\"\"6#%\"tGF*-%$sinG6#,&*&\"\"#F*F.F*F**&F)F*F4F,F,F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/2;" "6#/%!G,$*&\"\"\"F'\"\"#!\" \"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u[Pi/4](t)*cos*2*t;" "6#**-&%\"u G6#*&%#PiG\"\"\"\"\"%!\"\"6#%\"tGF*%$cosGF*\"\"#F*F.F*" }{TEXT -1 2 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = PIECEWISE( [0, t < Pi/4],[-cos*2*t/2, Pi/4 <= t]);" "6#/%!G-%*PIECEWISEG6$7$\"\"! 2%\"tG*&%#PiG\"\"\"\"\"%!\"\"7$,$**%$cosGF.\"\"#F.F+F.F5F0F01*&F-F.F/F 0F+" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Using " }{TEXT 0 10 "invlaplace" }{TEXT -1 11 " we obtai n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "exp(-Pi*s/4)/(s^2+4);\n`inverse transform`=inttrans[ invlaplace](%,s,t);\n'f(t)'=convert(rhs(%),piecewise);\nf := unapply(r hs(%),t):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*(\"\"%!\" \"%\"sG\"\"\"%#PiGF,F*F,,&*$)F+\"\"#F,F,F)F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2inverse~transformG,$*&#\"\"\"\"\"#F(*&-%*HeavisideG6 #,&%\"tGF(*&\"\"%!\"\"%#PiGF(F2F(-%$cosG6#,$*&F)F(F/F(F(F(F(F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$\"\"!2F ',$*&\"\"%!\"\"%#PiG\"\"\"F37$%*undefinedG/F'F.7$,$*&#F3\"\"#F3-%$cosG 6#,$*&F;F3F'F3F3F3F12F.F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The graph can be plotted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot(f(t),t= 0..2*Pi,-1..1,ytickmarks=3,thickness=2,labels=[t,`f(t)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 347 171 171 {PLOTDATA 2 "6'-%'CURVESG6$7dr7$$\"\" !F)F(7$$\"3i]cC&eb&p8!#=F(7$$\"3YqR*pd)>hDF-F(7$$\"3Zs[E^dK,RF-F(7$$\" 3ab^NehL]_F-F(7$$\"35*QhW&\\$Hf'F-F(7$$\"3]:5wGaJ:sF-F(7$$\"3zS11.fpPy F-F(7$$\"3#e!R^M[8#[)F-$\"3!)zEy49-li!#>7$$\"3upr'fwtl7*F-$\"3;bG_\"*o *)e7F-7$$\"3!Rbb2V`Iz*F-$\"3%G!f2\\*H3*=F-7$$\"3!QRa&4L&f/\"!#<$\"3EoH t@o@*[#F-7$$\"3_y%f^`(Q76FS$\"35/0O-^uTIF-7$$\"3YjXwg<#)y6FS$\"3R:nz\" H`1a$F-7$$\"3!or.w_drC\"FS$\"3?J$\\W.U'))RF-7$$\"36qGW%H$\\:8FS$\"3-Qp *=SUAO%F-7$$\"3SH22vMov8FS$\"3LdnJKh6CYF-7$$\"3$*)e)pbO(eV\"FS$\"3LWKp 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\"3YLMRuG)z)\\F-7$$\"3)=Zk[%R_$p%FS$\"3;^%>l_Tk*\\F-7$$\"3e;O;#eJ$4ZFS $\"3YTCXBl!***\\F-7$$\"3ns[!z6bes%FS$\"3g8/,Tn=)*\\F-7$$\"3lHhk`'yBu%F S$\"3s.x$4*)35*\\F-7$$\"3k'Q(Q*=-*eZFS$\"3//^^63Qy\\F-7$$\"3sU'G^sDax% FS$\"3#f)es#H;.'\\F-7$$\"3zb6h'zs%3[FS$\"3h2uleF'z!\\F-7$$\"3&)oO4o)>: %[FS$\"3:[rb(>CuRF-7$$\"3*zX#[4&eg 5&FS$\"3X%>q::l'GNF-7$$\"3#Q_\")pq87<&FS$\"3!*)))[3!HWQIF-7$$\"3n*e![/ *ojB&FS$\"3'pYL6V)p'\\#F-7$$\"3Dtq\\/&**HI&FS$\"3#4+H2wx*)*=F-7$$\"3#o b8X5I'p`FS$\"3g6wB!)Qen7F-7$$\"3fR3_p*3dV&FS$\"3s`/T/5)=>'FE7$$\"3PA\" GX$yy,bFS$!3\\&\\Yt0[1+%F\\w7$$\"3wxVy[u]ibFS$!3!oC[\"GE(RX'FE7$$\"39L 1/jqABcFS$!3+qaE\\F-7$$\"3mweKB,Ti dFS$!3Cf6fGITCDF-7$$\"3w\"4#\\qe;E`fFS$!3]XC[N#H/&RF-7$$\"3nKcu0ii>gFS$!3\\C bm^;I@VF-7$$\"3G:=>Xb9$3'FS$!35A)y\\6]^g%F-7$$\"3w)*zj%)[mYhFS$!3;PK'Q wwZ\"[F-7$$\"31*\\Lr)\\z!='FS$!3Ow[=x!Gb*[F-7$$\"3P***G'*3D\\@'FS$!3%Q :H0vxM&\\F-7$$\"3'*\\n(39!*>B'FS$!3+9^JRM\"Q(\\F-7$$\"3o*\\C@>b!\\iFS$ !3%*3QzueN))\\F-7$$\"3Q\\APV-7miFS$!3[;:l=\")3(*\\F-7$$\"3)****>YH&=$G 'FS$!3K)*************\\F--%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELS G6$%\"tG%%f(t)G-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VI EWG6$;F($\"+3`=$G'!\"*;$Fc`mF)$\"\"\"F)" 1 2 0 1 10 2 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 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 56 "Find the Laplace transforms of the follow ing functions. " }}{PARA 0 "" 0 "" {TEXT -1 11 " (a) " } {XPPEDIT 18 0 "f(t) = u[2](t)*(t-2)^3;" "6#/-%\"fG6#%\"tG*&-&%\"uG6#\" \"#6#F'\"\"\"*$,&F'F/F-!\"\"\"\"$F/" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "f(t) = u[Pi](t)*cos*t;" "6#/-%\"fG6#%\"tG*(-&%\"uG6#%#PiG6#F'\" \"\"%$cosGF/F'F/" }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "f(t)=u[1](t)* 3*exp(2-t)" "6#/-%\"fG6#%\"tG*(-&%\"uG6#\"\"\"6#F'F-\"\"$F--%$expG6#,& \"\"#F-F'!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 7 " (a) " }{XPPEDIT 18 0 "exp(-2*s)/s^4" "6#*&-%$expG6#,$*&\"\"#\"\" \"%\"sGF*!\"\"F**$F+\"\"%F," }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "-e xp(-s*Pi)*s/(s^2+1)" "6#,$*(-%$expG6#,$*&%\"sG\"\"\"%#PiGF+!\"\"F+F*F+ ,&*$F*\"\"#F+F+F+F-F-" }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "3/(s+1)* exp(1-s)" "6#*(\"\"$\"\"\",&%\"sGF%F%F%!\"\"-%$expG6#,&F%F%F'F(F%" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 51 "Find inverse Laplace t ransforms for the following. " }}{PARA 0 "" 0 "" {TEXT -1 11 " (a ) " }{XPPEDIT 18 0 "exp(-3*s)/s^2" "6#*&-%$expG6#,$*&\"\"$\"\"\"%\"sG F*!\"\"F**$F+\"\"#F," }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "exp(-Pi*s) /(s^2+9)" "6#*&-%$expG6#,$*&%#PiG\"\"\"%\"sGF*!\"\"F*,&*$F+\"\"#F*\"\" *F*F," }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "exp(-sqrt(2)*s)/(s+2)" " 6#*&-%$expG6#,$*&-%%sqrtG6#\"\"#\"\"\"%\"sGF-!\"\"F-,&F.F-F,F-F/" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 7 " (a) " } {XPPEDIT 18 0 "f(t) = PIECEWISE([0, t < 3],[t-3, 3 <= t])" "6#/-%\"fG6 #%\"tG-%*PIECEWISEG6$7$\"\"!2F'\"\"$7$,&F'\"\"\"F.!\"\"1F.F'" }{TEXT -1 10 " (b) " }{XPPEDIT 18 0 "f(t) = PIECEWISE([0, t <= Pi],[-sin *3*t/3, Pi < t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!1F'%#PiG7$,$ **%$sinG\"\"\"\"\"$F3F'F3F4!\"\"F52F.F'" }{TEXT -1 3 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " (c) " } {XPPEDIT 18 0 "f(t) = PIECEWISE([0, t < sqrt(2)],[exp(-2*t+2*sqrt(2)), sqrt(2) < t])" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"!2F'-%%sqrtG6#\" \"#7$-%$expG6#,&*&F1\"\"\"F'F8!\"\"*&F1F8-F/6#F1F8F82-F/6#F1F'" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 8 "Express " }{XPPEDIT 18 0 "f(t)=PIECEWISE([0,t<3],[t-3,t>=3])" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6 $7$\"\"!2F'\"\"$7$,&F'\"\"\"F.!\"\"1F.F'" }{TEXT -1 33 " in step func tion form and find " }{XPPEDIT 18 0 "L*[f(t)]" "6#*&%\"LG\"\"\"7#-%\"f G6#%\"tGF%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t)=u[3](t)*(t-3)" "6#/-%\"fG6#%\"tG*&-&%\"uG6#\"\"$6# F'\"\"\",&F'F/F-!\"\"F/" }{TEXT -1 7 ", so " }{XPPEDIT 18 0 "L*[f(t) ]=exp(-3*s)/s^2" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&*&-%$expG6#,$*&\"\" $F&%\"sGF&!\"\"F&*$F3\"\"#F4" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 8 "Express " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t-3, t < 3],[0, 3 <= t]);" "6#/-%\"fG6#%\"tG-%*PI ECEWISEG6$7$,&F'\"\"\"\"\"$!\"\"2F'F.7$\"\"!1F.F'" }{TEXT -1 33 " in \+ step function form and find " }{XPPEDIT 18 0 "L*[f(t)]" "6#*&%\"LG\"\" \"7#-%\"fG6#%\"tGF%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t-3-u[3](t)*(t-3);" "6#/-%\"fG6#%\"tG, (F'\"\"\"\"\"$!\"\"*&-&%\"uG6#F*6#F'F),&F'F)F*F+F)F+" }{TEXT -1 5 ", s o " }{XPPEDIT 18 0 "L*[f(t)]=1/(s^2)-3/s-exp(-3*s)/s^2" "6#/*&%\"LG\" \"\"7#-%\"fG6#%\"tGF&,(*&F&F&*$%\"sG\"\"#!\"\"F&*&\"\"$F&F/F1F1*&-%$ex pG6#,$*&F3F&F/F&F1F&*$F/F0F1F1" }{XPPEDIT 18 0 "`` = (1-exp(-3*s))/(s^ 2)-3/s;" "6#/%!G,&*&,&\"\"\"F(-%$expG6#,$*&\"\"$F(%\"sGF(!\"\"F0F(*$F/ \"\"#F0F(*&F.F(F/F0F0" }{TEXT -1 3 ". " }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "__ _____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " } }{PARA 0 "" 0 "" {TEXT -1 8 "Express " }{XPPEDIT 18 0 "f(t) = abs(t-3) " "6#/-%\"fG6#%\"tG-%$absG6#,&F'\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 "``=P IECEWISE([3-t, t < 3],[t-3, 3 <= t])" "6#/%!G-%*PIECEWISEG6$7$,&\"\"$ \"\"\"%\"tG!\"\"2F,F*7$,&F,F+F*F-1F*F," }{TEXT -1 34 " in step funct ion form and find " }{XPPEDIT 18 0 "L*[f(t)]" "6#*&%\"LG\"\"\"7#-%\"fG 6#%\"tGF%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(t) = 3-t+u[3](t)*(2*t-6);" "6#/-%\"fG6#%\"tG,(\"\"$\" \"\"F'!\"\"*&-&%\"uG6#F)6#F'F*,&*&\"\"#F*F'F*F*\"\"'F+F*F*" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "L*[f(t)] =3/s+(2*exp(-3*s)-1)/s^2" "6#/*&%\" LG\"\"\"7#-%\"fG6#%\"tGF&,&*&\"\"$F&%\"sG!\"\"F&*&,&*&\"\"#F&-%$expG6# ,$*&F.F&F/F&F0F&F&F&F0F&*$F/F4F0F&" }{TEXT -1 3 ". " }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 47 "_______________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 8 "Express " }{XPPEDIT 18 0 "f(t)= PIECEWISE([t, t < 1],[t-1, t<2],[1,t>=2])" "6#/-%\"fG6#%\" tG-%*PIECEWISEG6%7$F'2F'\"\"\"7$,&F'F-F-!\"\"2F'\"\"#7$F-1F2F'" } {TEXT -1 33 " in step function form and find " }{XPPEDIT 18 0 "L*[f(t )]" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t-u[1](t)+u[2](t )*(2-t)" "6#/-%\"fG6#%\"tG,(F'\"\"\"-&%\"uG6#F)6#F'!\"\"*&-&F,6#\"\"#6 #F'F),&F4F)F'F/F)F)" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "L*[f(t)]=(1-e xp(-2*s))/s^2-exp(-s)/s" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&,&*&,&F&F&- %$expG6#,$*&\"\"#F&%\"sGF&!\"\"F6F&*$F5F4F6F&*&-F06#,$F5F6F&F5F6F6" } {TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 47 "______________________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "Express " }{XPPEDIT 18 0 "f(t) = PIECEWISE([t, t < 1],[t-2, t < 3],[0 , 3 <= t]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6%7$F'2F'\"\"\"7$,&F'F-\"\" #!\"\"2F'\"\"$7$\"\"!1F3F'" }{TEXT -1 33 " in step function form and \+ find " }{XPPEDIT 18 0 "L*[f(t)]" "6#*&%\"LG\"\"\"7#-%\"fG6#%\"tGF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(t) = t-2*u[1](t)+u[3](t)*(2-t);" "6#/-%\"fG6#%\"tG,(F'\"\"\"*&\" \"#F)-&%\"uG6#F)6#F'F)!\"\"*&-&F.6#\"\"$6#F'F),&F+F)F'F1F)F)" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "L*[f(t)]=1/s^2-2*exp(-s)/s-exp(-3*s)*(1/s +1/s^2)" "6#/*&%\"LG\"\"\"7#-%\"fG6#%\"tGF&,(*&F&F&*$%\"sG\"\"#!\"\"F& *(F0F&-%$expG6#,$F/F1F&F/F1F1*&-F46#,$*&\"\"$F&F/F&F1F&,&*&F&F&F/F1F&* &F&F&*$F/F0F1F&F&F1" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 47 " _______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "__ _____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 14 "step func tion " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "p1 := plot([[[0,0],[2,0]],[[2,1],[4,1]]],thickness=2 ,color=red):\np2 := plot([[[2,0]],[[2,1]]$3],style=point,symbol=[circl e$2,diamond,cross],color=red):\nt1 := plots[textplot]([[2,-.13,`a`],[4 ,-.1,`t`],\n [-.2,1.5,`u (t)`]],\n \+ font=[HELVETICA,10],color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textpl ot]([-.23,1.45,`a`],\n font=[HELVETICA,9],color=COLOR(RGB,.0 1,.01,.01)):\nplots[display]([p1,p2,t1,t2],xtickmarks=0,\n ytic kmarks=3,view=[-.23..4,-.13..1.9]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "p1 := plot([[[-2,0],[0,0] ],[[0,1],[2,1]]],thickness=2,color=red):\np2 := plot([[[0,0]],[[0,1]]$ 3],style=point,\n symbol=[circle$2,diamond,cross],color=red):\n t1 := plots[textplot]([[2,-.1,`t`],\n [-.2,1.5,`H(t)`]] ,\n font=[HELVETICA,10],color=COLOR(RGB,.01,.01,.01 )):\nplots[display]([p1,p2,t1],xtickmarks=0,\n ytickmarks=3,vie w=[-2..2,-.13..1.9]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 27 "2nd shift formula examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 559 "p1 := plot(sin(t),t= Pi/2..14.14,thickness=2,color=red):\np2 := plot([[0,0],[Pi/2,0]],thick ness=2,color=red):\np3 := plot([[[Pi/2,0]],[[Pi/2,1]]$3],style=point, \n symbol=[circle$2,diamond,cross],color=red):\nt1 := plots[tex tplot]([[15,-.15,`t`]],font=[HELVETICA,10],\n color=COLOR(RGB,.01,.01 ,.01)):\nplots[display]([p1,p2,p3,t1],labels=[``,`f(t)`],\n xtickmark s=[0=`0`,1.57=`p/2`,3.14=`p`,4.71=`3p/2`,\n 6.28=`2p`,7.854=`5p/2`,9 .425=`3p`,11=`7p/2`,12.566=`4p`,14.14=`9p/2`],\n ytickmarks=3,font=[S YMBOL,10],labelfont=[HELVETICA,10],view=[0..15,-1.2..1.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 442 "f := t -> 24*exp(-t)*cos(2*t):\nvv := evalf(f(Pi)):\np1 := plot(f(t),t= Pi..8,thickness=2,color=red):\np2 := plot([[0,0],[Pi,0]],thickness=2,c olor=red):\np3 := plot([[[Pi,0]],[[Pi,vv]]$3],style=point,\n sy mbol=[circle$2,diamond,cross],color=red):\nt1 := plots[textplot]([[8.3 ,-.2,`t`]],font=[HELVETICA,10],\n color=COLOR(RGB,.01,.0 1,.01)):\nplots[display]([p1,p2,p3,t1],labels=[``,`f(t)`],\n tickmark s=[5,2],view=[0..8,-.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 29 "piecewise function examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "f := t -> 1 0*exp(-t):\nvv := evalf(f(2)):\np1 := plot(f(t),t=2..6,0..2,thickness= 2,color=red):\np2 := plot([[0,0],[2,0]],thickness=2,color=red):\np3 := plot([[[2,0]],[[2,vv]]$3],style=point,\n symbol=[circle$2,diam ond,cross],color=red):\nplots[display]([p1,p2,p3],tickmarks=[4,4],\n \+ view=[0..6,0..1.9],labels=[t,`f(t)`],ytickmarks=3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "f : = t -> 10*exp(-t):\nvv := evalf(f(2)):\np1 := plot(f(t),t=2..6,0..2,th ickness=2,color=red):\np2 := plot(t,t=0..2,0..2,thickness=2,color=red) :\np3 := plot([[[2,2]],[[2,vv]]$3],style=point,\n symbol=[circl e$2,diamond,cross],color=red):\nplots[display]([p1,p2,p3],tickmarks=[4 ,4],\n view=[0..6,0..2.2],labels=[t,`f(t)`],ytickmarks=3);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "summary picture " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 429 "p1 := plot(1-t^2/4,t=-1..0,color=red):\np2 := plot(1.5-t^2/4,t=0. .1,color=red):\np3 := plot([[[-1,0],[1,0]],[[0,0],[0,1.6]]],\n co lor=black,linestyle=[1,2]):\np4 := plot([[[0,1]],[[0,1.5]]$3],style=po int,\n symbol=[circle$2,diamond,cross],color=red):\nt1 := pl ots[textplot]([0,-.05,'a'],color=black):\nt2 := plots[textplot]([[-.55 ,1.06,'g(t)'],[.55,1.56,'h(t)']],color=red):\nplots[display]([p1,p2,p3 ,p4,t1,t2],axes=none);" }}}{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 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }