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{SECT 0 {EXCHG {PARA 219 "" 0 "" {TEXT 223 48 "Click on the plus sign \+
(+) to open the section. " }}}{SECT 1 {PARA 220 "" 0 "" {TEXT 230 83 "
1. Solving, and plotting solutions of Differential Equations: The \"ds
olve\" command." }}{PARA 221 "" 0 "" {TEXT 231 71 "Differential equati
ons and initial value problems are solved using the " }{TEXT 232 6 "ds
olve" }{TEXT 231 10 " command. " }}{PARA 221 "" 0 "" {TEXT 231 7 "usag
e: " }}{PARA 221 "" 0 "" {TEXT 232 7 "dsolve(" }{TEXT 233 1 " " }
{TEXT 231 25 "the differential equation" }{TEXT 233 1 " " }{TEXT 231 
24 ", the dependent variable" }{TEXT 233 4 ")   " }{TEXT 231 2 "or" }}
{PARA 221 "" 0 "" {TEXT 232 10 "dsolve( \{ " }{TEXT 231 45 "a differen
tial equation, initial condition(s)" }{TEXT 232 5 " \},  " }{TEXT 231 
22 "the dependent variable" }{TEXT 232 2 " )" }}{PARA 221 "" 0 "" }
{PARA 221 "" 0 "" {TEXT 231 31 "I start most sections with the " }
{TEXT 232 7 "restart" }{TEXT 231 55 " command to clear all previous va
riable definitions by " }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 8 
"restart;" }}}{PARA 221 "" 0 "" }{PARA 221 "" 0 "" {TEXT 232 9 "Exampl
e 1" }{TEXT 231 71 ": Find the general solution to the first order dif
ferential equation:  " }{TEXT 232 8 "y' = 3 y" }}{EXCHG {PARA 221 "> "
 0 "" {MPLTEXT 1 234 46 "sol1 := dsolve(diff(y(x),x) = 3 * y(x), y(x))
;" }{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%sol1G6
\"/-I\"yGF%6#I\"xGF%*&I$_C1GF%\"\"\"-I$expG6$I*protectedGF1I(_syslibGF
%6#,$F*\"\"$F-" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "" 0 "" {TEXT 
231 12 "Notice: The " }{XPPEDIT 18 0 "_C1;" "6#%$_C1G" }{TEXT 231 1 " \+
" }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 231 35 "  indicates an arbi
trary constant. " }}}{PARA 221 "" 0 "" }{PARA 221 "" 0 "" {TEXT 232 9 
"Example 2" }{TEXT 231 39 ":   Consider the initial value problem " }
{TEXT 232 8 "y' = 3 y" }{TEXT 231 5 " ,   " }{TEXT 232 12 "y(0) = 1/2.
 " }}{PARA 221 "" 0 "" {TEXT 231 1 "(" }{TEXT 232 1 "a" }{TEXT 231 21 
") Find the solution. " }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 
58 "sol2 := dsolve(\{diff(y(x),x) = 3*y(x), y(0) = 1/2\}, y(x));" }
{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%sol2G6\"/-
I\"yGF%6#I\"xGF%,$-I$expG6$I*protectedGF/I(_syslibGF%6#,$F*\"\"$#\"\"
\"\"\"#" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "" 0 "" {TEXT 231 94 "No
tice: Maple automatically solves for the arbitrary constant based on t
he initial condition. " }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 
27 "y2 := unapply(rhs(sol2),x);" }{MPLTEXT 1 234 1 "\n" }}{PARA 222 ""
 1 "" {XPPMATH 20 "6#>I#y2G6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%
,$-I$expG6$I*protectedGF0I(_syslibGF%6#,$9$\"\"$#\"\"\"\"\"#F%F%F%" }
{TEXT 235 1 " " }}}{EXCHG {PARA 221 "" 0 "" {TEXT 231 49 "You can plot
 the solution from x = 0 to x = 1 by " }}}{EXCHG {PARA 221 "> " 0 "" 
{MPLTEXT 1 234 84 "plot(y2(x),x=0..1,labels=[\"x\",\"y2(x)\"],title=\"
The solution to y' = 3y, y(0) = 1/2\");" }{MPLTEXT 1 234 1 "\n" }}
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$\"3pS:BCI-4%*Fdo7$$\"\"\"F)$\"3S$Qfh%oF/5!#;-%&COLORG6&%$RGBG$\"#5!\"
\"$F)F`[lFa[l-%&TITLEG6$QDThe~solution~to~y'~=~3y,~y(0)~=~1/26\"-%%FON
TG6$%*HELVETICAGF_[l-%+AXESLABELSG6$Q\"xFf[lQ&y2(x)Ff[lFg[l-%%VIEWG6$;
Fa[lF^[l;$\"2KB\"o2jW\"4$Fdo$\"1rD3$QiL-\"!#9" 1 2 2 0 10 1 2 6 1 4 2 
1.0 45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 221 "" 0 "" {TEXT 231 1 "
(" }{TEXT 232 1 "c" }{TEXT 231 23 ") Evaluate y2 at x = 2 " }}}{EXCHG 
{PARA 224 "> " 0 "" {MPLTEXT 1 236 6 "y2(2);" }}{PARA 222 "" 1 "" 
{XPPMATH 20 "6#,$-I$expG6$I*protectedGF'I(_syslibG6\"6#\"\"'#\"\"\"\"
\"#" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "" 0 "" {TEXT 231 53 "and ge
t a numerical value to 6 significant digits by " }}}{EXCHG {PARA 221 "
> " 0 "" {MPLTEXT 1 234 11 "evalf(%,6);" }{MPLTEXT 1 234 1 "\n" }}
{PARA 222 "" 1 "" {XPPMATH 20 "6#$\"'9<?!\"$" }{TEXT 235 1 " " }}}
{EXCHG {PARA 221 "> " 0 "" }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 
234 22 "solve(y2(x) = 100, x);" }{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 
1 "" {XPPMATH 20 "6#,$-I#lnG6$I*protectedGF'I(_syslibG6\"6#\"$+##\"\"
\"\"\"$" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 
234 11 "evalf(%,6);" }{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 1 "" 
{XPPMATH 20 "6#$\"'5m<!\"&" }{TEXT 235 1 " " }}}{PARA 221 "" 0 "" }
{PARA 221 "" 0 "" {TEXT 232 7 "Example" }{TEXT 231 1 " " }{TEXT 232 1 
"3" }{TEXT 231 50 ": Plot the solution to the differential equation: "
 }{TEXT 232 14 "t y' + 2 y = 2" }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"#"
 }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 231 3 "   "
 }{TEXT 232 9 "y(1) = 2." }}{PARA 221 "" 0 "" {TEXT 231 68 "To save on
 typing we can start by defining the differential equation" }}{EXCHG 
{PARA 221 "> " 0 "" {MPLTEXT 1 234 36 "ode3 := t*diff(y(t),t)+2*y(t)=2
*t^2;" }{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%od
e3G6\"/,&*&I\"tGF%\"\"\"-I%diffGI*protectedGF-6$-I\"yGF%6#F)F)F*F*F/\"
\"#,$*$F)F2F2" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "" 0 "" {TEXT 231 
28 "and then solve it using the " }{TEXT 232 6 "dsolve" }{TEXT 231 8 "
 command" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 36 "sol3 := dso
lve(\{ode3,y(1)=2\}, y(t));" }{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 1 "
" {XPPMATH 20 "6#>I%sol3G6\"/-I\"yGF%6#I\"tGF%*&,&*$F*\"\"%#\"\"\"\"\"
##\"\"$F1F0F0F*!\"#" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "> " 0 "" 
{MPLTEXT 1 234 27 "y3 := unapply(rhs(sol3),t);" }{MPLTEXT 1 234 1 "\n"
 }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I#y3G6\"f*6#I\"tGF%F%6$I)operator
GF%I&arrowGF%F%*&,&*$9$\"\"%#\"\"\"\"\"##\"\"$F3F2F2F/!\"#F%F%F%" }
{TEXT 235 1 " " }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 85 "plot(
y3(t),t=.5..3,labels=[\"t\",\"y3(t)\"],title=\"A plot of the solution \+
to Example 3\");" }{MPLTEXT 1 234 1 "\n" }}{PARA 223 "" 1 "" 
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#5!\"\"$Fi\\lFb]lFc]l-%&TITLEG6$QDA~plot~of~the~solution~to~Example~36
\"-%%FONTG6$%*HELVETICAGFa]l-%+AXESLABELSG6$Q\"tFh]lQ&y3(t)Fh]lFi]l-%%
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{TEXT 232 9 "Example 4" }{TEXT 231 64 ": Find and plot the solution to
 the initial value problem  y' = " }{XPPEDIT 18 0 "(3*x^2+4*x+2)/(2*(y
-1));" "6#*&,(*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*&\"\"%F'F)F'F'F*F'F'*&F*F'
,&%\"yGF'F'!\"\"F'F0" }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 231 1 "
 " }{TEXT 231 12 " , y(0)=-1. " }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 
1 234 54 "ode4 := diff(y(x),x) = (3*x^2 + 4*x + 2)/(2*(y(x)-1));" }
{MPLTEXT 1 234 1 "\n" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%ode4G6\"/-
I%diffGI*protectedGF)6$-I\"yGF%6#I\"xGF%F.*&,(*$F.\"\"#\"\"$F.\"\"%F2
\"\"\"F5,&F+F2!\"#F5!\"\"" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "> " 0
 "" {MPLTEXT 1 234 29 "dsolve(\{ode4,y(0)=-1\}, y(x));" }{MPLTEXT 1 
234 1 "\n" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#/-I\"yG6\"6#I\"xGF&,&\"
\"\"F**$,*\"\"%F**$F(\"\"$F**$F(\"\"#F1F(F1#F*F1!\"\"" }{TEXT 235 1 " \+
" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 24 "y4 := unapply(rhs(%
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G6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,&\"\"\"F-*$,*\"\"%F-*$9$
\"\"$F-*$F2\"\"#F5F2F5#F-F5!\"\"F%F%F%" }{TEXT 235 1 " " }}}{EXCHG 
{PARA 221 "> " 0 "" {MPLTEXT 1 234 82 "plot(y4(x),x=-3..1,labels=[\"x
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0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 221 "> "
 0 "" {MPLTEXT 1 234 16 "# end of section" }}}}{SECT 1 {PARA 220 "" 0 
"" {TEXT 230 69 "2. Plotting Direction Fields: The \"dfieldplot\" and \+
\"DEplot\" commands." }}{PARA 221 "" 0 "" {TEXT 232 45 "Logistic Growt
h: Example 1: x' = r x(1 - x/k)" }}{PARA 221 "" 0 "" {TEXT 231 62 "Fir
st we define the differential equation and give it a name. " }}{EXCHG 
{PARA 221 "> " 0 "" {MPLTEXT 1 234 8 "restart;" }}}{EXCHG {PARA 225 ">
 " 0 "" {MPLTEXT 1 237 7 "r := 1;" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#
>I\"rG6\"\"\"\"" }{TEXT 235 1 " " }}}{EXCHG {PARA 225 "> " 0 "" 
{MPLTEXT 1 237 7 "k := 5;" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I\"kG6
\"\"\"&" }{TEXT 235 1 " " }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 
234 48 "ode1 := diff(x(t),t) = r * x(t) * ( 1 -x(t)/k); " }{MPLTEXT 1 
234 1 "\n" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%ode1G6\"/-I%diffGI*pr
otectedGF)6$-I\"xGF%6#I\"tGF%F.*&F+\"\"\",&F0F0F+#!\"\"\"\"&F0" }
{TEXT 235 1 " " }}}{EXCHG {PARA 221 "" 0 "" {TEXT 231 65 "The command \+
for plotting direction fields is in a library called " }{TEXT 232 7 "D
Etools" }{TEXT 231 86 " in order to access this program we must first \+
load this library. To do this, use the " }{TEXT 233 4 "with" }{TEXT 
231 9 " command " }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 14 "wit
h(DEtools):" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 234 63 "dfieldpl
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