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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2280-1" }}{PARA 260 "" 0 "
" {TEXT -1 15 "March  20, 2006" }}{PARA 258 "" 0 "" {TEXT -1 58 "nonho
mogeneous linear systems of differential equations - " }}{PARA 261 "" 
0 "" {TEXT -1 24 " variation of parameters" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Work for example 4 page 363" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=matrix(2,2,[4,2,3,-1]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"%\"\"#7$\"
\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=matrix(2,1,
[7,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G-%'matrixG6#7$7#\"\"(
7#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f:=t->evalm(-t*e
xp(-2*t)*matrix(2,1,[15,4]));\n#the nonhomogeneous term." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%&eval
mG6#,$*(9$\"\"\"-%$expG6#,$*&\"\"#F2F1F2!\"\"F2-%'matrixG6%F8F27$\"#:
\"\"%F2F9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(s);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7#,$*(\"#:\"\"\"%\"sGF+-
%$expG6#,$*&\"\"#F+F,F+!\"\"F+F37#,$*(\"\"%F+F,F+F-F+F3" }}}{PARA 0 "
" 0 "" {TEXT -1 394 "Maple (or I) currently has trouble directly evalu
ating the solution formula: I can't seem to make Maple integrate matri
x valued functions without writing an explicit procedure (this did not
 used to be the case in older versions):  The procedure below takes a \+
matrix valued expression in s and returns a matrix expression in t, fo
r which each entry is the integral of the input from s=0 to s=t: " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
235 "G:=proc(mat,m,n,s)\nlocal H, #matrix function to be returned\n   \+
   i,  #row index\n      j;  #column index\n\nH:=matrix(m,n):\n for i \+
from 1 to m do\n   for j from 1 to n do\n   H[i,j]:=int(mat[i,j],s=0..
t):\n   od:\n od:\nreturn(evalm(H));\nend:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 117 "integrand:=evalm(exponential(-A,s)&*f(s));\n  #the
 integrand in the solution formula, page 4 notes or\n  #page 363 text
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*integrandG-%'matrixG6#7$7#,&**
\"#:\"\"\",&*&#\"\"'\"\"(F--%$expG6#,$*&\"\"&F-%\"sGF-!\"\"F-F-*&#F-F2
F--F46#,$*&\"\"#F-F9F-F-F-F-F-F9F--F46#,$*&FAF-F9F-F:F-F:**\"\"%F-,&*&
#FAF2F-F=F-F:*&#FAF2F-F3F-F-F-F9F-FBF-F:7#,&**F,F-,&*&#\"\"$F2F-F=F-F:
*&#FSF2F-F3F-F-F-F9F-FBF-F:**FGF-,&*&F<F-F3F-F-*&F0F-F=F-F-F-F9F-FBF-F
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sol:=t->evalm(exponent
ial(A,t)&*(x0+G(integrand,2,1,s)));\n   #solution formula" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$solGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%&ev
almG6#-%#&*G6$-%,exponentialG6$%\"AG9$,&%#x0G\"\"\"-%\"GG6&%*integrand
G\"\"#F9%\"sGF9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "si
mplify(sol(t));  #see page 364!!!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'matrixG6#7$7#,**&#\"\"$\"\"(\"\"\"-%$expG6#,$*&\"\"#F-%\"tGF-!\"\"F-
F-*&#F-F3F-*&F.F-)F4F3F-F-F5*&#\"#YF,F--F/6#,$*&\"\"&F-F4F-F-F-F-*(F3F
-F.F-F4F-F-7#,**&#\"#BF,F-F=F-F-*&F.F-F4F-F-*&#F3F,F-F.F-F5*&#F+F3F-F8
F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "16 0 0
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