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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Here is a worksheet exami
ning some Maple commands that are useful for checking your work in Cha
pter 6." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "We can perform partial fraction decomposition:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c
onvert((s+2)/(s^2-4*s+3),parfrac,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,&*&\"\"\"F%,&%\"sGF%F%!\"\"F(#!\"$\"\"#*&#\"\"&F+F%,&F'F%\"\"$F(F(F
%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
37 "We can perform the Laplace Transform:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "laplace(exp(2*t)*cos(3*t),t,
s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"sG\"\"\"\"\"#!\"\"F',&
*$)F%F(F'#F'\"\"*F'F'F)F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 36 "We can invert the Laplace Transform:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "invlaplace((s+1)/(s^2-4*s+5),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&-%$expG6#,$%\"tG\"\"#\"\"\"-%$cosG6#F)F+F+*(\"\"$F+F%F+-%$si
nGF.F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 95 "We can check our steps by manually solving a differential
 equation using the Laplace Transform:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "de:=diff(y(t),t$2)-2*di
ff(y(t),t)+2*y(t)=exp(-t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,
(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"*&F
1F2F*F2F2-%$expG6#,$F-F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"Y1:=laplace(de,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y1G/,,*&%
\"sG\"\"\",&*&F(F)-%(laplaceG6%-%\"yG6#%\"tGF2F(F)F)-F06#\"\"!!\"\"F)F
)--%\"DG6#F0F4F6*(\"\"#F)F(F)F,F)F6*&F<F)F3F)F)*&F<F)F,F)F)*&F)F),&F(F
)F)F)F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Y1A:=subs(\{y(0)
=0,D(y)(0)=1\},Y1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Y1AG/,**&)%
\"sG\"\"#\"\"\"-%(laplaceG6%-%\"yG6#%\"tGF2F)F+F+F+!\"\"*(F*F+F)F+F,F+
F3*&F*F+F,F+F+*&F+F+,&F)F+F+F+F3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "Y2:=solve(Y1A,laplace(y(t),t,s));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#Y2G*&,&%\"sG\"\"\"\"\"#F(F(,(*$)F'\"\"$F(F(*$)F'F)
F(!\"\"F)F(F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Y3:=conver
t(Y2,parfrac,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y3G,&*&\"\"\"F'
,&%\"sGF'F'F'!\"\"#F'\"\"&*&#F'F,F'*&,&!\")F'F)F'F',(*$)F)\"\"#F'F'*&F
5F'F)F'F*F5F'F*F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Y4:=
invlaplace(Y3,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Y4G,(-%$expG
6#,$%\"tG!\"\"#\"\"\"\"\"&*&#F-F.F-*&-F'6#F*F--%$cosGF3F-F-F+*(#\"\"(F
.F-F2F--%$sinGF3F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 62 "The Heaviside function is defined by:  U_c(t) =
 Heaviside(t-c)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "For example, plot U_1(t) - U_2(t)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(Heavis
ide(t-1)-Heaviside(t-2),t=0..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 423 
144 144 {PLOTDATA 2 "6%-%'CURVESG6$7co7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>
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#pa-NFhoF(7$$\"3!*******Rv&)zNFhoF(7$$\"3ILLLGUYoOFhoF(7$$\"3_mmm1^rZP
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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "f:=sin(t)*(Heaviside(t
)-Heaviside(t-Pi))+cos(t)*(Heaviside(t-Pi)-Heaviside(t-2*Pi));" }}
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