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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 
-1 23 "Solving a Linear System" }}{PARA 19 "" 0 "" {TEXT -1 8 "Math 30
8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "In t
his example, we see how to solve a linear system of the form  x'=Ax." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Suppose
" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "A = matrix([[1, -2], [4, -5]]);" 
"6#/%\"AG-%'matrixG6#7$7$\"\"\",$\"\"#!\"\"7$\"\"%,$\"\"&F-" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "and we want to solve" }}{PARA 
256 "" 0 "" {TEXT -1 6 "      " }{TEXT 256 1 "x" }{TEXT -1 5 "' = A" }
{TEXT 257 2 "x," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 258 1 "x
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[x(t)], [y(t)]]);" "6#-%'m
atrixG6#7$7#-%\"xG6#%\"tG7#-%\"yG6#F+" }}{PARA 0 "" 0 "" {TEXT -1 91 "
The differential equation written in matrix form above is really two f
irst order equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 20 "x'(t) = x(t) - 2y(t)" }}{PARA 256 "" 0 "" {TEXT -1 
21 "y'(t) = 4x(t) - 5y(t)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 72 "We define the problem in Maple by entering these e
quations individually:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "e
q1 := diff(x(t),t) = x(t)-2*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$eq1G/-%%diffG6$-%\"xG6#%\"tGF,,&F)\"\"\"*&\"\"#F.-%\"yGF+F.!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eq2 := diff(y(t),t) = 4*x(t
)-5*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/-%%diffG6$-%\"yG6
#%\"tGF,,&-%\"xGF+\"\"%*&\"\"&\"\"\"F)F3!\"\"" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "We can use the " }
{TEXT 259 7 "dsolve " }{TEXT -1 63 "function to solve the system.  As \+
usual, the first argument to " }{TEXT 260 6 "dsolve" }{TEXT -1 241 " i
s the problem to be solved, and the second argument holds the function
s to be found.  In this case, the problem is a system, so we put both \+
equations in brackets.  Also, there are two functions to be found, so \+
we also put them in brackets." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "sol := dsolve([eq1,eq2],[x(t),y(t)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$solG<$/-%\"yG6#%\"tG,&*&%$_C1G\"\"\"-%$expG6#,$F*!\"
$F.F.*&%$_C2GF.-F06#,$F*!\"\"F.F./-%\"xGF),&F,#F.\"\"#F4F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 345 "Note that Maple has returned the two fun
ctions in a list, enclosed in curly brackets.  To access the elements \+
of the list, we use the notation sol[i], where i is an integer.  It is
 important to also notice that Maple did not put x(t) first; the order
 tends to be random.  It might even be different the next time we ente
r the exact same command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 
"sol[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&%$_C1G\"
\"\"-%$expG6#,$F'!\"$F+#F+\"\"#*&%$_C2GF+-F-6#,$F'!\"\"F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol[1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG,&*&%$_C1G\"\"\"-%$expG6#,$F'!\"$F+F+*&%$
_C2GF+-F-6#,$F'!\"\"F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "The vector form of the solution would be
" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "matrix([[x(t)], [y(t)]]) = _C1*ma
trix([[1/2], [1]])*exp(-3*t)+_C2*matrix([[1], [1]])*exp(-t);" "6#/-%'m
atrixG6#7$7#-%\"xG6#%\"tG7#-%\"yG6#F,,&*(%$_C1G\"\"\"-F%6#7$7#*&F4F4\"
\"#!\"\"7#F4F4-%$expG6#,$*&\"\"$F4F,F4F;F4F4*(%$_C2GF4-F%6#7$7#F47#F4F
4-F>6#,$F,F;F4F4" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Check
 this--you should understand that this is the same solution as the ind
ividual functions in sol." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "If we ha
ve an initial value problem to solve, we must include the initial cond
itions as part of the problem to be solved in the " }{TEXT 261 6 "dsol
ve" }{TEXT -1 141 " function.  Suppse we want to solve the above syste
m, with the initial conditions x(0)=-1, y(0)=1.  Here is how we can us
e dsolve to do this:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sol
1 := dsolve([eq1,eq2,x(0)=-1,y(0)=1],[x(t),y(t)]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%sol1G<$/-%\"yG6#%\"tG,&-%$expG6#,$F*!\"\"!\"$*&\"
\"%\"\"\"-F-6#,$F*F1F4F4/-%\"xGF),&F,F1*&\"\"#F4F5F4F4" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 358 "Here is one way to pull apart Maple's so
lution and plot the results.  Note that when I executed these commands
, Maple put y(t) first and x(t) second in sol1.  If they are in a diff
erent order, you will have to change the indices in the following comm
ands.  Also note that I use the rhs function to get the right-hand sid
e of the expression returned by Maple." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "solx := rhs(sol1[2]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%solxG,&-%$expG6#,$%\"tG!\"\"!\"$*&\"\"#\"\"\"-F'6#,$F*F,F/F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "soly := rhs(sol1[1]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solyG,&-%$expG6#,$%\"tG!\"\"!\"$*
&\"\"%\"\"\"-F'6#,$F*F,F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "plot(\{solx,soly\},t=-0.25..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 333 
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 60 "You can tell which curve is which by the initia
l conditions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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