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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 258 74 "Linear ODE with Constan
t Coefficients\nMethod of Undetermined Coefficients " }}{PARA 257 "" 
0 "" {TEXT 256 47 "Date: Feb 20, 2002\nLast Revision: Feb 20, 2002\n" 
}{TEXT 267 7 "Maple 6" }}{PARA 259 "" 0 "" {TEXT 259 16 "Bent E. Peter
sen" }}{PARA 258 "" 0 "" {TEXT 260 17 "bent@alum.mit.edu" }}{PARA 258 
"" 0 "" {TEXT 261 22 "petersen@math.orst.edu" }}{PARA 0 "" 0 "" {TEXT 
262 0 "" }}{PARA 0 "" 0 "" {TEXT 263 15 "Course: Mth 256" }}{PARA 0 "
" 0 "" {TEXT 264 17 "Term: Winter 2002" }}{PARA 0 "" 0 "" {TEXT 265 
11 "File name: " }{TEXT 257 32 "256w2002-undetermined-coeffs.mws" }
{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 557 "For each of the following Cauchy initial value problems \+
do the following: (1) Find the complementary solution, that is, a gene
ral solution of the associated homogeneous equation; (2) Use the metho
d of undetermined coefficients to find a particular solution of the in
homogeneous equation; (3)  Combine the results of the first two parts \+
to find a general solution of the inhomogeneous equation;  (4) Determi
ne the values of the parameters (arbitrary constants) in the general s
olution found in the previous step so as to satisfy the initial values
 condition." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 47 "Does your solution agree with Maple's solution?" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 1
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "ode01:=diff(y(x),x,x)+3*diff(y(x),x)+2*y(x)=x^2+1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode01G/,(-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F-\"\"#\"\"\"*&\"\"$F2-F(6$F*F-F2F2*&F1F2F*F2F2,&*$)F-F1F2F2F2F2
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "init01:=y(0)=3,D(y)(0)=
-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init01G6$/-%\"yG6#\"\"!\"\"$
/--%\"DG6#F(F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsol
ve(\{ode01,init01\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6
#%\"xG,,*$)F'\"\"#\"\"\"#F,F+*&#\"\"$F+F,F'F,!\"\"#\"\"*\"\"%F,*&#\"\"
&F4F,-%$expG6#,$F'!\"#F,F1*&F+F,-F96#,$F'F1F,F," }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 2" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ode
02:=diff(y(x),x,x)-diff(y(x),x)=1+exp(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&ode02G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"
\"-F(6$F*F-!\"\",&F2F2-%$expGF,F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "init02:=y(0)=1,D(y)(0)=2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'init02G6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)\"\"#" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(\{ode02,init02\},y(x
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*F'!\"\"*&-%$exp
GF&\"\"\"F'F-F-*&\"\"#F-F+F-F-F-F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 3" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode03:=diff
(y(x),x,x)+4*y(x)=x*cos(2*x)+x*cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&ode03G/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F
2F*F2F2,&*&F-F2-%$cosG6#,$F-F1F2F2*&F-F2-F8F,F2F2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "init03:=y(0)=A,D(y)(0)=B;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'init03G6$/-%\"yG6#\"\"!%\"AG/--%\"DG6#F(F)%\"BG" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(\{ode03,init03\},y(
x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,.*&,&#!$p\"\"$w
&\"\"\"*&#F.\"\"#F.%\"BGF.F.F.-%$sinG6#,$F'F1F.F.*&%\"AGF.-%$cosGF5F.F
.*&,&#!\"\"\"#kF.*&#F.\"\")F.)F'F1F.F.F.F3F.F.*(#F.\"\"$F.F'F.-F:F&F.F
.*(#F.\"#;F.F'F.F9F.F.*&#F1\"\"*F.-F4F&F.F." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 4" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ode04:
=diff(y(x),x,x)+2*diff(y(x),x)+10*y(x)=exp(-x)+exp(-x)*sin(3*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode04G/,(-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"#5F2F*F2F2,&-%$expG6#,$F-!\"
\"F2*&F9F2-%$sinG6#,$F-\"\"$F2F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "init04:=y(0)=1,D(y)(0)=-2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'init04G6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)!\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(\{ode04,init04\},y(x)
); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*-%$expG6#,$F'!
\"\"#\"\"\"\"\"**&#\"\"&\"#=F/*&F)F/-%$sinG6#,$F'\"\"$F/F/F-*&#F/\"\"'
F/*(F)F/-%$cosGF8F/F'F/F/F-*(#\"\")F0F/F)F/F?F/F/" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 5" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "o
de05:=diff(y(x),x,x)-4*y(x)=exp(2*x)-exp(-2*x)+exp(x)-exp(-x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode05G/,&-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2!\"\",*-%$expG6#,$F-F1F2-F86#,$F-!\"#F
5-F8F,F2-F86#,$F-F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "in
it05:=y(0)=0,D(y)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init05G6
$/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "dsolve(\{ode05,init05\},y(x)): simplify(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,.-%$expG6#,$F'\"\"##\"\"\"\"#
C*(#F/\"\"%F/-F*6#,$F'!\"#F/F'F/F/*&#F/\"\"$F/-F*F&F/!\"\"*&#F/F:F/-F*
6#,$F'F<F/F/*(F2F/F'F/F)F/F/*&#F/F0F/F4F/F<" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 6" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ode06:
=diff(y(x),x$3)+3*diff(y(x),x$2)+3*diff(y(x),x)+y(x)=x+x*exp(-x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode06G/,*-%%diffG6$-%\"yG6#%\"xG-%
\"$G6$F-\"\"$\"\"\"*&F1F2-F(6$F*-F/6$F-\"\"#F2F2*&F1F2-F(6$F*F-F2F2F*F
2,&F-F2*&F-F2-%$expG6#,$F-!\"\"F2F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "init06:=y(0)=2,D(y)(0)=-1,(D@@2)(y)(0)=2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'init06G6%/-%\"yG6#\"\"!\"\"#/--%\"DG6#F(F
)!\"\"/---%#@@G6$F/F+F0F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "dsolve(\{ode06,init06\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"yG6#%\"xG,.F'\"\"\"\"\"$!\"\"*(#F)\"#CF)-%$expG6#,$F'F+F))F'\"\"%
F)F)*&\"\"&F)F/F)F)*(F*F)F'F)F/F)F)*(#F*\"\"#F)F/F))F'F:F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Pro
blem 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "ode07:=diff(y(x),x$4)-y(x)=x+exp(x)-exp(-x)+cos(x)-si
n(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode07G/,&-%%diffG6$-%\"yG6
#%\"xG-%\"$G6$F-\"\"%\"\"\"F*!\"\",,F-F2-%$expGF,F2-F66#,$F-F3F3-%$cos
GF,F2-%$sinGF,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "init07:
=y(0)=1,D(y)(0)=-3,(D@@2)(y)(0)=2,(D@@3)(y)(0)=-1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'init07G6&/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)!\"$/--
-%#@@G6$F/\"\"#F0F)F8/---F66$F/\"\"$F0F)!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "dsolve(\{ode07,init07\},y(x)): simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,4-%$sinGF&#\"\"\"\"\"#*
&#F,F-F,-%$expGF&F,!\"\"*&#\"\"$\"\"%F,-%$cosGF&F,F2*&#\"\"*F6F,-F16#,
$F'F2F,F,*&#F,F6F,*&F)F,F'F,F,F2*&#F,F6F,*&F'F,F7F,F,F2*(#F,F6F,F'F,F<
F,F,F'F2*(FFF,F'F,F0F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "Problem 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "ode08:=diff(y(x),x$3)-6
*diff(y(x),x$2)-9*diff(y(x),x)+14*y(x)=exp(x)+exp(-x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&ode08G/,*-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"$
\"\"\"*&\"\"'F2-F(6$F*-F/6$F-\"\"#F2!\"\"*&\"\"*F2-F(6$F*F-F2F:*&\"#9F
2F*F2F2,&-%$expGF,F2-FC6#,$F-F:F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "init08:=y(log(2))=0,D(y)(log(2))=0,(D@@2)(y)(log(2))=
0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init08G6%/-%\"yG6#-%#lnG6#\"
\"#\"\"!/--%\"DG6#F(F)F./---%#@@G6$F2F-F3F)F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "dsolve(\{ode08,init08\},y(x));" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&,(*&-%$expG6#,$F'\"\"*\"\"\"F'F1#!
\"\"\"#=*&#F1\"#;F1-F-6#,$F'\"\"(F1F1*&#F1\"$3\"F1F,F1F1F1-F-6#,$F'!\"
)F1F1*&,&-%#lnG6#\"\"##F1F4#F1\"$W\"F3F1-F-F&F1F1*&#\"#9\"#\")F1-F-6#,
$F'!\"#F1F3*&#\"#>\"'w<LF1F8F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Problem 9" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ode09:=diff(y(x),
x$2)-5*diff(y(x),x)+6*y(x)=x^2*exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&ode09G/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"&F
2-F(6$F*F-F2!\"\"*&\"\"'F2F*F2F2*&)F-F1F2-%$expGF,F2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "init09:=y(0)=2,D(y)(0)=3;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'init09G6$/-%\"yG6#\"\"!\"\"#/--%\"DG6#F(F)\"\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(\{ode09,init09\}
,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,-%$expGF&#
\"\"(\"\"%*(#\"\"$\"\"#\"\"\"F'F2F)F2F2*(#F2F1F2)F'F1F2F)F2F2-F*6#,$F'
F1F2*&#F0F-F2-F*6#,$F'F0F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 10 "Problem 10" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ode10:=diff(y(x),
x$2)+y(x)=sin(x)+cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ode10G/
,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*F2,&-%$sinGF,F2-%$cosG
F,F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "init10:=y(0)=A,D(y)
(0)=B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'init10G6$/-%\"yG6#\"\"!%
\"AG/--%\"DG6#F(F)%\"BG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "
dsolve(\{ode10,init10\},y(x)): simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,,-%$sinGF&#\"\"\"\"\"#*&F)F,%\"BGF,F,*&-
%$cosGF&F,%\"AGF,F,*(F+F,F)F,F'F,F,*&#F,F-F,*&F'F,F1F,F,!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "40 1" 0 }
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