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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 259 48 "Variation of Parameters
 and\nDuhamel's Principle " }}{PARA 257 "" 0 "" {TEXT 257 47 "Date: Fe
b 26, 2002\nLast Revision: Feb 26, 2002\n" }{TEXT 268 7 "Maple 6" }}
{PARA 259 "" 0 "" {TEXT 260 16 "Bent E. Petersen" }}{PARA 258 "" 0 "" 
{TEXT 261 17 "bent@alum.mit.edu" }}{PARA 258 "" 0 "" {TEXT 262 22 "pet
ersen@math.orst.edu" }}{PARA 0 "" 0 "" {TEXT 263 0 "" }}{PARA 0 "" 0 "
" {TEXT 264 15 "Course: Mth 256" }}{PARA 0 "" 0 "" {TEXT 265 17 "Term:
 Winter 2002" }}{PARA 0 "" 0 "" {TEXT 266 11 "File name: " }{TEXT 258 
28 "256w2002_varparm_duhamel.mws" }{TEXT 267 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "
In this worksheet we illustrate Duhamel's principle and variation of p
arameters. We use Maple to ease the calculations though, of course, if
 one is going to use Maple there is generally little reason to do vari
ation of parameters. Our reason here is pedagogical." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "We consider only secon
d order linear differential equations. Corresponding results hold for \+
higher order." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 74 "Consider a linear inhomogeneous ordinary differential \+
equation of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 55 "ode00:=diff(y(t),t,t)+p(t)*diff(y(t),t)+q
(t)*y(t)=g(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode00G/,(-%%diffG
6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&-%\"pGF,F2-F(6$F*F-F2F2*&-%\"qGF
,F2F*F2F2-%\"gGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 91 "The method of variation of parameters gives us a p
articular solution of  ode00  in the form" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "y(t)=Int(K(t,s)*g(s)
,s=a..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG-%$IntG6$*&
-%\"KG6$F'%\"sG\"\"\"-%\"gG6#F/F0/F/;%\"aGF'" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "K(t,s)=(y
1(s)*y2(t)-y1(t)*y2(s))/(y1(s)*D(y2)(s)-D(y1)(s)*y2(s));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"KG6$%\"tG%\"sG*&,&*&-%#y1G6#F(\"\"\"-%#y2G6
#F'F/F/*&-F-F2F/-F1F.F/!\"\"F/,&*&F,F/--%\"DG6#F1F.F/F/*&--F;6#F-F.F/F
5F/F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "and  y1, y2  is a fundamental solution set for the associ
ated homogeneous equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 164 "For each  s  we see  K(t,s)  is a linear combina
tion of  y1(t)  and  y2(t)  and so is a solution of the associated lin
ear homogeneous ordinary differential equation" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ode00h:=lhs
(ode00)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ode00hG/,(-%%diffG6$-
%\"yG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&-%\"pGF,F2-F(6$F*F-F2F2*&-%\"qGF,F2
F*F2F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 56 "Moreover, it is clearly the solution with initial value
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "y(s)=0,D(y)(s)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
/-%\"yG6#%\"sG\"\"!/--%\"DG6#F%F&\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "This characterization of \+
 K(t,s) as the solution (for each  s) of an initial value problem for \+
a " }{TEXT 256 11 "homogeneous" }{TEXT -1 305 " linear ordinary differ
ential equation,  and the corresponding result for higher order equati
ons, is known as Duhamel's principle. Note while  K(t,s)  is defined i
n terms of a fundamental solution set, Duhamel's principle shows us th
at it doesn't actually depend on the choice of fundamental solution se
t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Duh
amel's principle has a physical interpretation. It also has theoretica
l applications." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "Here then is a procedure to calculate  K(t,s):" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d
uhamel:=proc(ode,y,t,s)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  local \+
odeh,inith,solnh,K;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "odeh:=lhs(od
e)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "inith:=y(s)=0,D(y)(s)=1;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "K:=rhs(dsolve(\{odeh,inith\},y(t)
));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Note this procedure i
s not very robust. We make no error checking and we make strong assump
tions about the way our differential equation is written." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let's look at some
 examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 269 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "ode01:=(x-1)*diff(y(x),x,x)-x*diff(
y(x),x)+y(x)=(x-1)^2/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode01G/,
(*&,&%\"xG\"\"\"F*!\"\"F*-%%diffG6$-%\"yG6#F)-%\"$G6$F)\"\"#F*F**&F)F*
-F-6$F/F)F*F+F/F**&*$)F(F5F*F*F)F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "K01:=duhamel(ode01,y,x,s);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$K01G,&*&%\"xG\"\"\",&%\"sGF(F(!\"\"F+F+*&*&F*F(-%$ex
pG6#F'F(F(*&-F/6#F*F(F)F(F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "g01:=subs(x=s,rhs(ode01)/(x-1));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$g01G*&,&%\"sG\"\"\"F(!\"\"F(F'F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "int(K01*g01,s): y01:=simplify(subs(s=x,%));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y01G,&*&%\"xG\"\"\"-%#lnG6#F'F(!\"
\"F(F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "Let's compare our result with Maple's solution of  ode01.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "dsolve(ode01,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%\"yG6#%\"xG,**&F'\"\"\"-%#lnGF&F*!\"\"F*F-*&%$_C1GF*F'F*F**&%$_C2
GF*-%$expGF&F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 128 "Setting the parameters  _C1 and _C2 to 0 we a par
ticular solution and it is the same as the one we found by Duhamel's p
rinciple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 270 10 "Example 02" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ode02:=diff(y(x),x,x)-y(x)=exp(2*x)
/(1+exp(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode02G/,&-%%diffG6$
-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*!\"\"*&-%$expG6#,$F-F1F2,&F2F2-F6F
,F2F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "K02:=duhamel(ode02
,y,x,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$K02G,&*&-%$expG6#%\"xG
\"\"\"-F(6#%\"sG!\"\"#F+\"\"#*&#F+F1F+*&F,F+-F(6#,$F*F/F+F+F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g02:=subs(x=s,rhs(ode02));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g02G*&-%$expG6#,$%\"sG\"\"#\"\"\"
,&F,F,-F'6#F*F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int
(K02*g02,s): y02:=simplify(subs(s=x,%));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$y02G,**&-%$expG6#%\"xG\"\"\"-%#lnG6#,&F+F+F'F+F+#F+\"\"#*&#F+
\"\"%F+F'F+!\"\"F0F+*&#F+F1F+*&-F(6#,$F*F5F+F,F+F+F5" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Again let's com
pare our result with Maple's solution of  ode02." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(ode0
2,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,.*&-%$expGF
&\"\"\"-%#lnG6#,&F,F,F*F,F,#F,\"\"#*&#F,\"\"%F,F*F,!\"\"F1F,*&#F,F2F,*
&-F+6#,$F'F6F,F-F,F,F6*&%$_C1GF,F*F,F,*&%$_C2GF,F:F,F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 10 "Example \+
03" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "ode03:=diff(y(t),t,t)+y(t)=tan(t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&ode03G/,&-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"\"#\"
\"\"F*F2-%$tanGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "K03:=d
uhamel(ode03,y,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$K03G,&*&-%$
cosG6#%\"sG\"\"\"-%$sinG6#%\"tGF+F+*&-F-F)F+-F(F.F+!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g03:=subs(t=s,rhs(ode03));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g03G-%$tanG6#%\"sG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(K03*g03,s): y03:=simplify(subs(
s=t,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y03G,$*&-%$cosG6#%\"tG
\"\"\"-%#lnG6#*&,&F+F+-%$sinGF)F+F+F'!\"\"F+F3" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Again let's compare \+
our result with Maple's solution of  ode03." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(ode03,y(t)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,(*&-%$cosGF&\"\"
\"-%#lnG6#,&-%$secGF&F,-%$tanGF&F,F,!\"\"*&%$_C1GF,-%$sinGF&F,F,*&%$_C
2GF,F*F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "Again we have agreement." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 
10 "Example 04" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 71 "ode04:=x^2*diff(y(x),x,x)+x*diff(y(x),x)+(x^2-
1/4)*y(x)=x^(5/2)*cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode04G
/,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F)-%\"$G6$F)F*F+F+*&F)F+-F-6$F
/F)F+F+*&,&*$F(F+F+#F+\"\"%!\"\"F+F/F+F+*&)F)#\"\"&F*F+-%$cosGF1F+" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "K04:=duhamel(ode04,y,x,s);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$K04G,&*&*(-%%sqrtG6#%\"sG\"\"\"
-%$cosG6#F+F,-%$sinG6#%\"xGF,F,*$-F)6#F3F,!\"\"F,*&*(F(F,-F1F/F,-F.F2F
,F,*$-F)6#F3F,F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g04:=
subs(x=s,rhs(ode04)/x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g04G*&
-%%sqrtG6#%\"sG\"\"\"-%$cosG6#F)F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "int(K04*g04,s): y04:=simplify(subs(s=x,%));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$y04G,$*&-%%sqrtG6#%\"xG\"\"\",&-%$cosG6#F
*F+*&-%$sinGF/F+F*F+F+F+#F+\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Again let's compare our result wit
h Maple's solution of  ode04." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(ode04,y(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&*&%$_C1G\"\"\"-%$cos
GF&F,F,*$-%%sqrtG6#F'F,!\"\"F,*&*&%$_C2GF,-%$sinGF&F,F,*$-F16#F'F,F3F,
*(#F,\"\"%F,-F16#F'F,F-F,F,*(F=F,)F'#\"\"$\"\"#F,F7F,F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Agreement
 again!" }}}}{MARK "46 1 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }