{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "# This demo shows how to u
se maple to solve higher order linear differential equations." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" 
{TEXT -1 115 "# Example 1. Consider the second order initial value pro
blem given by y\" + 2y' + y =x^2 + 1 + e^x, y(0)=0, y'(0)=2." }}{PARA 
0 "" 0 "" {TEXT -1 61 "# The next two steps show how to define the pro
blem by maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 66 "diffeq1:=diff(y(x),x$2) + 2*diff(y(x),x) + y(x) \+
= x^2 +1 + exp(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "inits
1:=y(0)=0, D(y)(0)=2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "# To get
 the general solution to the differential equation, we use:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol1g:=dsolve(diffeq1,y(x));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "# Note that the solution abo
ve involves two constants, namely _C1 and _C2. We can solve _C1 and _C
2 by our initial conditions. " }}{PARA 0 "" 0 "" {TEXT -1 45 "# To sol
ve the initial value problem, we use:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "dsolve(\{diffeq1,inits1\}, y(x));" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 57 "# If you don't like the output format above, you \+
may try:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "expand(\"); sol1p:=simplify(\",exp);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 87 "# To plot the solution of the initial val
ued problem on the interval [-1,3], we may use" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(rhs(sol1p
), x=-1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "# To obtain a fl
oating decimal value for the solution at x=2.5, we may use the command
 \"subs\" and \"evalf\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"subs(x=2.5, sol1p);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval
f(\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "# Example 2. Consider \+
a third order initial valued problem, y''' + y\" + 3y'-5y=2+6x-5x^2, y
(0)=-1, y'(0)=1, y'''(0)=-3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "#
 Once again, we first define the differential equation and the initial
 conditions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "diffeq2:=di
ff(y(x),x$3) + diff(y(x), x$2) + 3*diff(y(x),x) - 5*y(x) = 2+6*x-5*x^2
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "inits2:=y(0)=-1, D(y)(
0)=1, (D@@2)(y)(0)=-3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "# Then \+
we solve the initial value problem:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "dsolve(\{diffeq2,inits2\},y(x));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 47 "# If you prefer a different form of the answer:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "expand(\");sol2:=simplify(
\",exp);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "# Example 3. Not eve
ry initial value problem can be solved by dsolve. Here is an example t
hat we cannot get the solution by dsolve. " }}{PARA 0 "" 0 "" {TEXT 
-1 50 "# We have to approximate the solution numerically." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "# First, d
efine the initial value problem:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "diffeq3:=t^2*diff(y(t),t$2)-y(t)*diff(y(t),t)=y(t)^(-
3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "inits3:=y(1)=0.5, D(
y)(1)=2.1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "# Solve the initia
l valued problem numerically, and pay attention to the key word \"nume
ric\" in the dsolve command.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "sol3:=dsolve(\{diffeq3,inits3\},y(t),numeric);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 63 "# If you want the approximation value of the so
lution at x=1.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sol3(1.
5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "# If you want to plot the
 graph of the numerical solution on the interval [0.6, 1.2], then do t
he following steps:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with
(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "odeplot(sol3,[t
,y(t)],0.6..1.2);" }}}}{MARK "34 0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 }