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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 31 "Solving a Differential Eq
uation" }}{PARA 18 "" 0 "" {TEXT -1 21 "Math 374    Fall 1999" }}
{PARA 19 "" 0 "" {TEXT -1 31 "Jay Treiman,  Spetember 7, 1999" }}
{PARA 19 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "
It is always a good idea to " }{HYPERLNK 17 "restart" 2 "restart" "" }
{TEXT -1 22 " Maple before starting" }}{PARA 0 "" 0 "" {TEXT -1 56 "a \+
new worksheet.  This guarantees that no variables are " }}{PARA 0 "" 
0 "" {TEXT -1 16 "assigned values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 33 "Inputting a differential equation" }}{PARA 0 "" 0 "" 
{TEXT -1 56 "In Maple one needs to explicitly specify which variables
" }}{PARA 0 "" 0 "" {TEXT -1 59 "in a differential equation are functi
ons of other variable." }}{PARA 0 "" 0 "" {TEXT -1 41 "For example, in
 the differential equation" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "diff(y,x) = x-y;" "6#/-%%diffG6$%\"yG%\"xG,&F(\"\"\"F'!
\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 32 "it is necessary t
o specify that " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 18 " is a fun
ction of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Here is the differen
tial equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "de1 := diff(y(x),x)= x -y(x);" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 42 "Having Maple solve a differential equatio
n" }}{PARA 0 "" 0 "" {TEXT -1 30 "Maple has a built in routine, " }
{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 26 ", for soling differe
ntial " }}{PARA 0 "" 0 "" {TEXT -1 62 "equations.  Looking at the help
 file for dsolve will give you " }}{PARA 0 "" 0 "" {TEXT -1 64 "the fo
rmat.  The first argument is a differential equation or a " }}{PARA 0 
"" 0 "" {TEXT -1 67 "set of differential equations and the second argu
ment is a function" }}{PARA 0 "" 0 "" {TEXT -1 52 "or set of functions
 for which one is solving the DE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 55 "Here Maple solves the differential equati
on from above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "gsoln := dsolve(de1,y(x));" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "One can substitute the so
lution back into the original differential " }}{PARA 0 "" 0 "" {TEXT 
-1 19 "equation using the " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT 
-1 37 " command and then evaluate the result" }}{PARA 0 "" 0 "" {TEXT 
-1 44 "to see if both sides are the same using the " }{HYPERLNK 17 "ev
al" 2 "eval" "" }{TEXT -1 4 " or " }{HYPERLNK 17 "evalb" 2 "evalb" "" 
}{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solved := subs(gsoln,de1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(solved);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalb(solved);" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 31 "Integrating to find a solution." }}{PARA 0 "" 
0 "" {TEXT -1 46 "Since Maple will integrate many functions, it " }}
{PARA 0 "" 0 "" {TEXT -1 47 "is easy to solve many differential equati
ons of" }}{PARA 0 "" 0 "" {TEXT -1 8 "the form" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "diff(y(x),x) = f(x);" "6#/-%%diffG6$-%
\"yG6#%\"xGF*-%\"fG6#F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
51 " Here is the soltuion oto the differential equation" }}{PARA 259 "
" 0 "" {XPPEDIT 18 0 "diff(x(t),`$`(t,2)) = g;" "6#/-%%diffG6$-%\"xG6#
%\"tG-%\"$G6$F*\"\"#%\"gG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v(t) := int(g,t)+C1;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x(t) := int(v(t),t) + C
2;" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Plotting slope fields and
 solution curves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 55 "Maple has a set of routines that help when working with" 
}}{PARA 0 "" 0 "" {TEXT -1 54 "differential equations.  They are in a \+
package called " }{HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 23 "First load the package." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(
DEtools):" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 46 "The routine one uses to plot a slope field and" }}{PARA 0 "" 0 
"" {TEXT -1 24 "particular solutions is " }{HYPERLNK 17 "DEplot" 2 "DE
tools,DEplot" "" }{TEXT -1 20 ".  One can use it to" }}{PARA 0 "" 0 "
" {TEXT -1 21 "plot the slope field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "DEplot(\{de1\},\{y(x)\},x=
-5..5,y=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "One can also add particular solutions by adding a " }}
{PARA 0 "" 0 "" {TEXT -1 51 "list of initial conditions.  Check the op
tions for " }{HYPERLNK 17 "DEplot" 2 "DEtools,DEplot" "" }{TEXT -1 3 "
 to" }}{PARA 0 "" 0 "" {TEXT -1 50 "learn about adding a title or chan
ging the colors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 117 "DEplot(\{de1\},\{y(x)\},x=-5..5,\n     [[y(0)
=3],[y(0)=-4],[y(-3)=0]],\n     y=-5..5,linecolor=blue,title=`diff(y,x
)=x-y`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "Find the solution to the differential equation" }}{PARA 258 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(x,t) = t^2-x;" "6#/-%%diffG6$
%\"xG%\"tG,&*$F(\"\"#\"\"\"F'!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "using Maple.  Then check your answer in Maple." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Find the solution to " }}{PARA 
260 "" 0 "" {XPPEDIT 18 0 "diff(y(x),x,x) = 4*x-cos(x);" "6#/-%%diffG6
%-%\"yG6#%\"xGF*F*,&*&\"\"%\"\"\"F*F.F.-%$cosG6#F*!\"\"" }{TEXT -1 1 "
 " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 26 " using
 Maple to integrate." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Graph the
 solutions of  " }{XPPEDIT 18 0 "diff(y(x),x)-y = x^2;" "6#/,&-%%diffG
6$-%\"yG6#%\"xGF+\"\"\"F)!\"\"*$F+\"\"#" }{TEXT -1 20 "  simultaneousl
y for" }}{PARA 0 "" 0 "" {TEXT -1 24 "the innitial conditions " }
{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "y(0) = -1;" "6#/-%\"yG6#\"\"!,$\"\"\"!\"\"" }{TEXT -1 
6 ", and " }{XPPEDIT 18 0 "y(1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 
1 "." }}}}}{MARK "0 0 0" 31 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }