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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 26 "Laplace Transform in Ma
ple" }}{PARA 0 "" 0 "" {TEXT 256 20 "Mth 256 March 7 2001" }}{PARA 0 "
" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 35 
"Filename: 256winter2001_laplace.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "with(inttrans): with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "In addit
ion to just computing Laplace and inverse Laplace transforms, Maple ca
n apply the Laplace transform directly to a linear differential equati
on:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "ode1:=diff(y(t),t,t)+4*y(t)=cos(t);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%ode1G/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"
\"\"F*\"\"%-%$cosGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Lap
1:=laplace(ode1,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Lap1G/,(*&
%\"sG\"\"\",&*&F(\"\"\"-%(laplaceG6%-%\"yG6#%\"tGF3F(F,F,-F16#\"\"!!\"
\"F,F,--%\"DG6#F1F5F7F-\"\"%*&F(F),&*$)F(\"\"#F)F,F,F,!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We
 can make the equation look comfortably familiar by introducing " }
{XPPEDIT 18 0 "Y(t);" "6#-%\"YG6#%\"tG" }{TEXT -1 52 "  for the Laplac
e transform of the unknown function " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG
6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Lp1:=subs(laplace(y(t),t,s)=Y(s),La
p1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Lp1G/,(*&%\"sG\"\"\",&*&F(
\"\"\"-%\"YG6#F(F,F,-%\"yG6#\"\"!!\"\"F,F,--%\"DG6#F1F2F4F-\"\"%*&F(F)
,&*$)F(\"\"#F)F,F,F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "Now we let Maple do the algebra" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s
olve(Lp1,Y(s)): Y(s):=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"YG6#%
\"sG*&,,*&-%\"yG6#\"\"!\"\"\")F'\"\"$\"\"\"F/*&F'F/F+F2F/*&--%\"DG6#F,
F-F/)F'\"\"#F2F/F5F/F'F/F2,(*$)F'\"\"%F2F/*$F9F2\"\"&F>F/!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "To
 solve the differential equation we can now take the inverse Laplace t
ransform" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "soln1:=invlaplace(Y(s),s,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&soln1G,*-%$cosG6#,$%\"tG\"\"##!\"\"\"\"$*&-%\"yG6#\"
\"!\"\"\"F&F4F4*&--%\"DG6#F1F2F4-%$sinGF(F4#F4F+-F'6#F*#F4F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "No
tice the solutions are conveniently paramterized by the initial values
  " }{XPPEDIT 18 0 "y(0);" "6#-%\"yG6#\"\"!" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "D(y)(0);" "6#--%\"DG6#%\"yG6#\"\"!" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Of course
, if we just want to solve the differential equation we can do so dire
ctly:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "init1:=y(0)=A,D(y)(0)=B;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&init1G6$/-%\"yG6#\"\"!%\"AG/--%\"DG6#F(F)%\"BG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "soln1b:=dsolve(\{ode1,init1
\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'soln1bG/-%\"yG6#%\"tG,
**&,&-%$sinGF(#\"\"\"\"\"%-F.6#,$F)\"\"$#F0\"#7F0-F.6#,$F)\"\"#F0F0*&,
&-%$cosGF3F6-F?F(F/F0-F?F9F0F0*&%\"BGF0F8\"\"\"#F0F;*&,&#!\"\"F5F0%\"A
GF0F0FAFDF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "subs(y(0)=A,
D(y)(0)=B,soln1)-rhs(soln1b): simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 57 "In spite of initial appearances the two solutio
ns agree! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 166 "The nice thing about the Laplace transform of course is that i
t correctly handles the case where the driving term has discontinuitie
s. Let's look at a simple example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ode2:=diff(y(t),t,t)-diff(y
(t),t)+6*y(t)=Heaviside(t-1)-Heaviside(t-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ode2G/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"
-F(6$F*F-!\"\"F*\"\"',&-%*HeavisideG6#,&F-F2F5F2F2-F96#,&F-F2!\"#F2F5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Lap2:=laplace(ode2,t,s)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Lap2G/,,*&%\"sG\"\"\",&*&F(\"
\"\"-%(laplaceG6%-%\"yG6#%\"tGF3F(F,F,-F16#\"\"!!\"\"F,F,--%\"DG6#F1F5
F7F+F7F4F,F-\"\"',&*&-%$expG6#,$F(F7F)F(!\"\"F,*&-F@6#,$F(!\"#F)F(FCF7
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
22 "Before we substitute  " }{XPPEDIT 18 0 "Y(s);" "6#-%\"YG6#%\"sG" }
{TEXT -1 115 "  we had best unassign it. Otherwise we will be substitu
ting its previous value rather than just a new symbol for  " }
{XPPEDIT 18 0 "laplace(y(t),t,s);" "6#-%(laplaceG6%-%\"yG6#%\"tGF)%\"s
G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 17 "unassign('Y(s)');" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "Lp2:=subs(laplace(y(t),t,s)=Y(s),Lap2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$Lp2G/,,*&%\"sG\"\"\",&*&F(\"\"\"-%\"YG6#F
(F,F,-%\"yG6#\"\"!!\"\"F,F,--%\"DG6#F1F2F4F+F4F0F,F-\"\"',&*&-%$expG6#
,$F(F4F)F(!\"\"F,*&-F=6#,$F(!\"#F)F(F@F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "solve(Lp2,Y(s)): Y(s):=%;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"YG6#%\"sG*&,,*&)F'\"\"#\"\"\"-%\"yG6#\"\"!\"\"\"F2
*&--%\"DG6#F/F0F2F'F2F2*&F'F-F.F-!\"\"-%$expG6#,$F'F9F2-F;6#,$F'!\"#F9
F-*&F'\"\"\",(*$F+F-F2F'F9\"\"'F2\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "soln2:=invlaplace(Y(s),s,t);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%&soln2G,4**-%\"yG6#\"\"!\"\"\"-%$expG6#,$%\"tG#F+\"
\"#F+-%%sqrtG6#\"#B\"\"\"-%$sinG6#,$*&F3F7F0F+F1F+#!\"\"F6*(F'F7F,F7-%
$cosGF:F+F+**--%\"DG6#F(F)F+F,F7F3F7F8F7#F2F6-%*HeavisideG6#,&F0F+F>F+
#F+\"\"'**FHF+-F-6#,&F0F1#F>F2F+F+F3F7-F96#,$*&F3F7FKF+F1F+#F+\"$Q\"*(
FHF7FOF7-FAFTF+#F>FM-FI6#,&F0F+!\"#F+Fen**FfnF+-F-6#,&F0F1F>F+F+F3F7-F
96#,$*&F3F7FhnF+F1F+#F>FX*(FfnF7F[oF7-FAF_oF+FL" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "exmp2:=subs(y(0)=0,D(y)(0)=0,soln2);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%&exmp2G,.-%*HeavisideG6#,&%\"tG\"\"\"!\"\"
F+#F+\"\"'**F&F+-%$expG6#,&F*#F+\"\"##F,F5F+F+-%%sqrtG6#\"#B\"\"\"-%$s
inG6#,$*&F7F;F)F+F4F+#F+\"$Q\"*(F&F;F0F;-%$cosGF>F+#F,F.-F'6#,&F*F+!\"
#F+FF**FGF+-F16#,&F*F4F,F+F+F7F;-F=6#,$*&F7F;FIF+F4F+#F,FB*(FGF;FLF;-F
EFPF+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(exmp2,t=0..
3,thickness=3,color=red);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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7,Hl?Fen$\"1&yC$[S\\OVF07$$\"1+v$4')QY4#Fen$\"1)*yIffHAWF07$$\"1+]P4w)
R7#Fen$\"1%3+]#)pu[%F07$$\"1+vVVN2c@Fen$\"1:IgiG*R`%F07$$\"1++]x%f\")=
#Fen$\"1=LFlme`XF07$$\"1+v$4%)\\$=AFen$\"17%Q<<Nla%F07$$\"1+]P/-a[AFen
$\"1eP2gg.9XF07$$\"1+]7Ly4!G#Fen$\"1],CvN@_WF07$$\"1+](=Yb;J#Fen$\"1y2
y;\\RhVF07$$\"1++]i@OtBFen$\"10qo+#Q()4%F07$$\"1+]PfL'zV#Fen$\"1(y$*op
oDq$F07$$\"1+++!*>=+DFen$\"12ug]ih0KF07$$\"1++DE&4Qc#Fen$\"1'y/:(G<&e#
F07$$\"1+]P%>5pi#Fen$\"1g&f+6zc'=F07$$\"1+vou;!fl#Fen$\"1!=tHjGK]\"F07
$$\"1+++bJ*[o#Fen$\"1=*)H8yUA6F07$$\"1+]7j17=FFen$\"1ZtF#G4Cl'F,7$$\"1
++Dr\"[8v#Fen$\"1KwRyFH\")=F,7$$\"1+]i]s1\"y#Fen$!1WhXK^9MDF,7$$\"1+++
Ijy5GFen$!1V:?N.:pqF,7$$\"1+v=nIZUGFen$!1[_%eUq7?\"F07$$\"1+]P/)fT(GFe
n$!1()3Ba@>/<F07$$\"1+++b!)[/HFen$!17!Gs6\"*4>#F07$$\"1+]i0j\"[$HFen$!
1())o.u30o#F07$$\"1+D\"G:3u'HFen$!16Kr@Wb1KF07$$\"\"$F($!1at4V^EHPF0-%
'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#Fi[l-%+AXESLABELSG6$Q
\"t6\"%!G-%%VIEWG6$;F(Fh[l%(DEFAULTG" 1 2 0 1 3 2 6 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 75 "Be sure to check the Maple help facility \+
for more information and examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 114 "Note as mentioned at the beginning Maple
 can also be used just to compute transforms. Here's a couple of examp
les:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "laplace(exp(at)*sin(omega*t-phi),t,s);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&-%$expG6#%#atG\"\"\",&*&*&-%$cosG6#%$phiGF)%
&omegaGF)\"\"\",&*$)%\"sG\"\"#F2F)*$)F1F7F2F)!\"\"!\"\"*&*&-%$sinGF/F)
F6F)F2F3F:F)F)F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "invlapl
ace((2*s-3)/((s^2+4)^2*(s+1)),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,,-%$expG6#,$%\"tG!\"\"#F)\"\"&-%$cosG6#,$F(\"\"##\"\"\"F+-%$sinGF.#!
\"$\"#!)*&F(F2F,F2#F)\"\")*&F(\"\"\"F3F2#F2\"\"%" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "That sure beats do
ing the algebra by hand!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "28 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }