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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 62 "Computer Problem #3: An e
lectrical circuit with a periodic emf" }}{PARA 19 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 211 "Below is a brief introduction to usi
ng Maple to do Laplace transform problems. Included is the solution to
 p. 349: Problem 47 . Your assignment is to use Maple to do p. 349: Pr
oblem 48. Graph your solution from " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"
\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=10" "6#/%\"tG\"#5" }{TEXT 
-1 5 " for " }{XPPEDIT 18 0 "L=1" "6#/%\"LG\"\"\"" }{TEXT -1 11 " henr
y and " }{XPPEDIT 18 0 "R=1,2" "6$/%\"RG\"\"\"\"\"#" }{TEXT -1 5 " and
 " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 6 " ohms." }}}{EXCHG {PARA 
4 "" 0 "" {TEXT -1 26 "Laplace transform basics: " }}{PARA 0 "" 0 "" 
{TEXT -1 30 "Laplace transforms are in the " }{TEXT 256 8 "inttrans" }
{TEXT -1 9 " package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-%)addtableG%(fourierG%+fouriercosG%
+fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhilbertG%+invlaplaceG
%(laplaceG%'mellinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "lapl
ace(t^2,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$%\"sG!\"$\"\"#" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(%,s,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%\"tG\"\"#" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 61 "Hence to solve a differential equation by Laplace trans
forms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "diffeq := D(y)(t)
-y(t) = exp(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'diffeqG/,&--%\"D
G6#%\"yG6#%\"tG\"\"\"-F+F,!\"\"-%$expGF," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "laplace(diffeq,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/,(*&%\"sG\"\"\"-%(laplaceG6%-%\"yG6#%\"tGF.F&F'F'-F,6#\"\"!!\"\"F(
F2*$,&F&F'F2F'F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Apply the ini
tial condition, say " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"
" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(y(
0)=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"sG\"\"\"-%(laplace
G6%-%\"yG6#%\"tGF.F&F'F'!\"\"F'F(F/*$,&F&F'F/F'F/" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "solve(%,laplace(y(t),t,s));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&%\"sG\"\"\",(*$F$\"\"#F%F$!\"#F%F%!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(%,s,t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*&%\"tG\"\"\"-%$expG6#F%F&F&F'F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "is the solution." }}{PARA 4 "" 0 "
" {TEXT -1 24 "A more difficult problem" }}{PARA 0 "" 0 "" {TEXT -1 
70 "We now do problem 46, p. 303, which is the LR series circuit equat
ion " }{XPPEDIT 18 0 "L*diff(i(t),t)+R*i(t)=E(t)" "6#/,&*&%\"LG\"\"\"-
%%diffG6$-%\"iG6#%\"tGF.F'F'*&%\"RGF'-F,6#F.F'F'-%\"EG6#F." }{TEXT -1 
20 ", where the voltage " }{XPPEDIT 18 0 "E(t)" "6#-%\"EG6#%\"tG" }
{TEXT -1 48 " is the periodic function shown in Figure 7.48. " }{TEXT 
-1 0 "" }{TEXT -1 70 "First we use the formula for Laplace transform o
f a periodic function " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "L(f(t)) = i
nt(f(t)*exp(-s*t),t=0..T)/(1-exp(-s*T))" "6#/-%\"LG6#-%\"fG6#%\"tG*&-%
$intG6$*&-F(6#F*\"\"\"-%$expG6#,$*&%\"sGF2F*F2!\"\"F2/F*;\"\"!%\"TGF2,
&\"\"\"F2-F46#,$*&F8F2F=F2F9F9F9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "f := unapply(piecewise(t<=1,1,-1),t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piecewi
seG6%19$\"\"\"F1!\"\"F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "int(exp(-s*t)*f(t),t=0..2)/(1-exp(-2*s));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&*(,&\"\"\"F'-%$expG6#%\"sG!\"#F'F+!\"\"-F)6#,$F+F,F
'F'*$F+F-F'F',&F'F'F.F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "LTE := simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LTEG*(,&
-%$expG6#%\"sG\"\"\"!\"\"F+F+,&F'F+F+F+F,F*F," }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 63 "Thus the Laplace transform of the full differential eq
uation is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "laplace(L*D(i)
(t)+R*i(t),t,s) = LTE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"LG\"
\"\",&*&%\"sGF'-%(laplaceG6%-%\"iG6#%\"tGF1F*F'F'-F/6#\"\"!!\"\"F'F'*&
%\"RGF'F+F'F'*(,&-%$expG6#F*F'F5F'F',&F:F'F'F'F5F*F5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "subs(i(0)=0,%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*(%\"LG\"\"\"%\"sGF'-%(laplaceG6%-%\"iG6#%\"tGF/F(F'
F'*&%\"RGF'F)F'F'*(,&-%$expG6#F(F'!\"\"F'F',&F4F'F'F'F7F(F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Iinv := solve(%,laplace(i(t)
,t,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%IinvG*(,&-%$expG6#%\"sG
\"\"\"!\"\"F+F+F*F,,**(%\"LGF+F*F+F'F+F+*&F/F+F*F+F+*&%\"RGF+F'F+F+F2F
+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Let's try to get Maple to \+
do this one directly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "inv
laplace(Iinv,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%+invlaplaceG
6%*(%\"sG!\"\",**(%\"LG\"\"\"F(F--%$expG6#F(F-F-*&F,F-F(F-F-*&%\"RGF-F
.F-F-F3F-F)F.F-F(%\"tGF--F%6%*&F(F)F*F)F(F4F)" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 70 "Nope didn't work. So let's try to do it by expanding i
nto a series in " }{XPPEDIT 18 0 "exp(-s)" "6#-%$expG6#,$%\"sG!\"\"" }
{TEXT -1 20 " . We first replace " }{XPPEDIT 18 0 "exp(-s)" "6#-%$expG
6#,$%\"sG!\"\"" }{TEXT -1 28 " by a simpler variable. Let " }{XPPEDIT 
18 0 "r=exp(-s)" "6#/%\"rG-%$expG6#,$%\"sG!\"\"" }{TEXT -1 18 " which \+
means that " }{XPPEDIT 18 0 "exp(s)=1/r" "6#/-%$expG6#%\"sG*&\"\"\"\"
\"\"%\"rG!\"\"" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "subs(exp(s)=1/r,Iinv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*$
%\"rG!\"\"\"\"\"F'F(F(%\"sGF',**(%\"LGF(F)F(F&F'F(*&F,F(F)F(F(*&%\"RGF
(F&F'F(F/F(F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Since the denomi
nator of Iinv involves " }{XPPEDIT 18 0 "exp(-s)" "6#-%$expG6#,$%\"sG!
\"\"" }{TEXT -1 12 " the answer " }{XPPEDIT 18 0 "i(t)" "6#-%\"iG6#%\"
tG" }{TEXT -1 78 " will change formulas every 1 second. Thus if we wan
t to plot the answer from " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "t=10" "6#/%\"tG\"#5" }{TEXT -1 27 " \+
we need the first 10 terms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "series(%,r=0,10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"rG*&%\"s
G!\"\",&*&%\"LG\"\"\"F&F+F+%\"RGF+F'\"\"!,$F%!\"#\"\"\",$F%\"\"#\"\"#F
.\"\"$F1\"\"%F.\"\"&F1\"\"'F.\"\"(F1\"\")F.\"\"*-%\"OG6#F+\"#5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Now we eliminate the " }{XPPEDIT 
18 0 "O(t^10)" "6#-%\"OG6#*$%\"tG\"#5" }{TEXT -1 18 " term and replace
 " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "exp
(-s)" "6#-%$expG6#,$%\"sG!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "convert(%,polynom);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,6*&%\"sG!\"\",&*&%\"LG\"\"\"F%F*F*%\"RGF*F&F**(F%F&F'F
&%\"rGF*!\"#*(F%F&F'F&F-\"\"#F0*(F%F&F'F&F-\"\"$F.*(F%F&F'F&F-\"\"%F0*
(F%F&F'F&F-\"\"&F.*(F%F&F'F&F-\"\"'F0*(F%F&F'F&F-\"\"(F.*(F%F&F'F&F-\"
\")F0*(F%F&F'F&F-\"\"*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"iinv2 := subs(r=exp(-s),%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&iin
v2G,6*&%\"sG!\"\",&*&%\"LG\"\"\"F'F,F,%\"RGF,F(F,*(F'F(F)F(-%$expG6#,$
F'F(F,!\"#*(F'F(F)F(F/\"\"#F5*(F'F(F)F(F/\"\"$F3*(F'F(F)F(F/\"\"%F5*(F
'F(F)F(F/\"\"&F3*(F'F(F)F(F/\"\"'F5*(F'F(F)F(F/\"\"(F3*(F'F(F)F(F/\"\"
)F5*(F'F(F)F(F/\"\"*F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Now Map
le can invert it" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ansLR :
= invlaplace(iinv2,s,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ansLRG,
J*$%\"RG!\"\"\"\"\"*&F'F(-%$expG6#,$*(F'F)%\"tGF)%\"LGF(F(F)F(*&-%*Hea
visideG6#,&F0F)F(F)F)F'F(!\"#*(F3F)F'F(-F,6#,$*(F'F)F6F)F1F(F(F)\"\"#*
&-F46#,&F0F)F7F)F)F'F(F=*(F?F)F'F(-F,6#,$*(F'F)FAF)F1F(F(F)F7*&-F46#,&
F0F)!\"$F)F)F'F(F7*(FHF)F'F(-F,6#,$*(F'F)FJF)F1F(F(F)F=*&-F46#,&F0F)!
\"%F)F)F'F(F=*(FRF)F'F(-F,6#,$*(F'F)FTF)F1F(F(F)F7*&-F46#,&F0F)!\"&F)F
)F'F(F7*(FfnF)F'F(-F,6#,$*(F'F)FhnF)F1F(F(F)F=*&-F46#,&F0F)!\"'F)F)F'F
(F=*(F`oF)F'F(-F,6#,$*(F'F)FboF)F1F(F(F)F7*&-F46#,&F0F)!\"(F)F)F'F(F7*
(FjoF)F'F(-F,6#,$*(F'F)F\\pF)F1F(F(F)F=*&-F46#,&F0F)!\")F)F)F'F(F=*(Fd
pF)F'F(-F,6#,$*(F'F)FfpF)F1F(F(F)F7*&-F46#,&F0F)!\"*F)F)F'F(F7*(F^qF)F
'F(-F,6#,$*(F'F)F`qF)F1F(F(F)F=" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
18 0 "Heaviside(t)" "6#-%*HeavisideG6#%\"tG" }{TEXT -1 50 "is an alter
native name for the unit step function " }{XPPEDIT 18 0 "U(t)" "6#-%\"
UG6#%\"tG" }{TEXT -1 36 ". We convert ansLR to a function of " }
{XPPEDIT 18 0 "t,L" "6$%\"tG%\"LG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 
"R" "6#%\"RG" }{TEXT -1 5 " and " }{TEXT -1 0 "" }{TEXT -1 24 "graph t
he solution with " }{XPPEDIT 18 0 "L=1" "6#/%\"LG\"\"\"" }{TEXT -1 11 
" henry and " }{XPPEDIT 18 0 "R=4" "6#/%\"RG\"\"%" }{TEXT -1 6 " ohms:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ians := unapply(ansLR,t
,L,R);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%iansGR6%%\"tG%\"LG%\"RG6
\"6$%)operatorG%&arrowGF*,J*$9&!\"\"\"\"\"*&F0F1-%$expG6#,$*(F0F29$F29
%F1F1F2F1*&-%*HeavisideG6#,&F9F2F1F2F2F0F1!\"#*(F<F2F0F1-F56#,$*(F0F2F
?F2F:F1F1F2\"\"#*&-F=6#,&F9F2F@F2F2F0F1FF*(FHF2F0F1-F56#,$*(F0F2FJF2F:
F1F1F2F@*&-F=6#,&F9F2!\"$F2F2F0F1F@*(FQF2F0F1-F56#,$*(F0F2FSF2F:F1F1F2
FF*&-F=6#,&F9F2!\"%F2F2F0F1FF*(FenF2F0F1-F56#,$*(F0F2FgnF2F:F1F1F2F@*&
-F=6#,&F9F2!\"&F2F2F0F1F@*(F_oF2F0F1-F56#,$*(F0F2FaoF2F:F1F1F2FF*&-F=6
#,&F9F2!\"'F2F2F0F1FF*(FioF2F0F1-F56#,$*(F0F2F[pF2F:F1F1F2F@*&-F=6#,&F
9F2!\"(F2F2F0F1F@*(FcpF2F0F1-F56#,$*(F0F2FepF2F:F1F1F2FF*&-F=6#,&F9F2!
\")F2F2F0F1FF*(F]qF2F0F1-F56#,$*(F0F2F_qF2F:F1F1F2F@*&-F=6#,&F9F2!\"*F
2F2F0F1F@*(FgqF2F0F1-F56#,$*(F0F2FiqF2F:F1F1F2FFF*F*6\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(ians(t,1,4),t=0..10);" }}
{PARA 13 "" 1 "" {GLPLOT2D 318 318 318 {PLOTDATA 2 "6%-%'CURVESG6$7]x7
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