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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "# In this demo, we are  go
ing to learn how to solve a system of homogeneous equations by maple.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "# Example 1. The first example will show
 how to solve  the second order homogeneous equation  y\" + y' + y =0:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "# Define the differential equ
ation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "diffeq:=diff(y(t)
,t$2)+diff(y(t),t)+y(t)=0;" }{TEXT -1 0 "" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 48 "# Use dsolve to solve the differential eq
uation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sol:=dsolve(\",y
(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "# Observe that the solu
tion is a linear combination of the two fundamental solutions. We set \+
_C1= 1 and _C2=0 to extract the first " }}{PARA 0 "" 0 "" {TEXT -1 91 
"# fundamental solution. Then set _C1=0 and _C2=1 to extract the other
 fundemantal solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y
1:=rhs(subs(\{_C2=0,_C1=1\},sol));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "y2:=rhs(subs(\{_C2=1,_C1=0\},sol));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "# Exam
ple 2. For higher order equations, dsolve is not adequate. We are now \+
going to solve  y'''' + y' + y =0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 53 "# First define the forth order differential equation:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "diffeq:=diff(y(t),t$4)+diff(
y(t),t)+y(t)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sol:=dso
lve(\",y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "# The summation
 we got above is over all the complex roots of _Z^4 + _Z  + 1 = 0. So,
 to get the fundamental set of real solutions, we" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "# need to solve _Z^4  +  _Z  + 1 = 0 first." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ev:=fsolve(r^4+r+1=0,r,complex);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "# Then substitute each of the co
mplex root into e^(rt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
Y1:=evalc(exp(ev[1]*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"Y2:=evalc(exp(ev[2]*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "Y3:=evalc(exp(ev[3]*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "Y4:=evalc(exp(ev[4]*t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
134 "# We then observe that Y1 and Y2, Y3 and Y4 form two conjugate pa
irs. To obtain the fundamental set of real solutions, we may take the
" }}{PARA 0 "" 0 "" {TEXT -1 34 "# linear combination of Y1 and Y2:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "y1:=(Y1+Y2)/2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y2:=(Y2-Y1)/(2*I);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 90 "# Or we may use Re and Im to extract the \+
real and imaginary parts of a complex expression." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "y3:=evalc(Re(Y3));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "y4:=evalc(Im(Y4));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 93 "# y1, y2, y3 and y4 then form a fundamental set of real s
olutions to our 4th order equation. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }