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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 91 "Define a second order differential equation:  Note especially h
ow we define the exponential" }}{PARA 0 "" 0 "" {TEXT -1 89 "function,
 and use multiplication signs.  This is problem 23, page 178.  We will
 solve the" }}{PARA 0 "" 0 "" {TEXT -1 93 "general differential equati
on, then solve an Initial Value Problem (IVP), and finally we will" }}
{PARA 0 "" 0 "" {TEXT -1 16 "plot the result." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "de:=diff(y(
t),t$2)-4*diff(y(t),t)+4*y(t)=2*t^2+4*t*exp(2*t)+t*sin(2*t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F-\"
\"#\"\"\"*&\"\"%F2-F(6$F*F-F2!\"\"*&F4F2F*F2F2,(*&F1F2)F-F1F2F2*(F4F2F
-F2-%$expG6#,$*&F1F2F-F2F2F2F2*&F-F2-%$sinGF?F2F2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "A:=dsolve(de,y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG/-%\"yG6#%\"tG,2*&-%$expG6#,$*&\"\"#\"\"\"F)F2F2F
2%$_C2GF2F2*(F)F2F,F2%$_C1GF2F2*&#F2\"#[F2*&,&*&\"\"'F2F)F2F2\"\"$F2F2
-%$cosGF.F2F2F2*&F1!\"\"F)F1F2F)F2#F=\"\"%F2*&#F1F=F2*&)F)F=F2F,F2F2F2
*&#F2\"#;F2-%$sinGF.F2FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 100 "This is the general solution- we can pic
k out the solution to the homogeneous part- we can see there" }}{PARA 
0 "" 0 "" {TEXT -1 101 "was a double root of r=2,   the particular par
t of the solution corresponding to t^2 was t^2/2+t+3/4," }}{PARA 0 "" 
0 "" {TEXT -1 99 "the particular part corresponding to 4te^(2t) is 2/3
t^3e^(2t)  (Notice that it is a t^3- why?), and" }}{PARA 0 "" 0 "" 
{TEXT -1 65 "finally for tsin(2t), we have (6t+3)/48 * cos(2t)- (1/16)
*sin(2t)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
46 "We'll now solve an IVP, with y(0)=3, y'(0)=-2:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "B:=dsolve(
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\"BG/-%\"yG6#%\"tG,2*&#\"#N\"#;\"\"\"-%$expG6#,$*&\"\"#F/F)F/F/F/F/*&#
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.F/-%$sinGF2F/F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(rh
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