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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 73 "Lab 1: Direction Fields,
 Solving First Order ODE's and Plotting Solutions" }}{PARA 257 "" 0 "
" {TEXT -1 42 "Click on the + symbol to open that section" }}{PARA 0 "
" 0 "" {TEXT 272 78 "Evaluate all input (in red) by placing the cursor
 anywhere in red and enter.  " }{TEXT -1 1 " " }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 62 "1. Solving, and plotting solutions of,  Differential
 Equations" }{TEXT 260 23 ": The \"dsolve\" command." }}{PARA 0 "" 0 "
" {TEXT -1 71 "Differential equations and initial value problems are s
olved using the " }{TEXT 271 6 "dsolve" }{TEXT -1 10 " command. " }}
{PARA 0 "" 0 "" {TEXT -1 7 "usage: " }}{PARA 0 "" 0 "" {TEXT 257 7 "ds
olve(" }{TEXT -1 0 "" }{TEXT 258 1 " " }{TEXT -1 25 "the differential \+
equation" }{TEXT 266 1 " " }{TEXT -1 24 ", the dependent variable" }
{TEXT 267 4 ")   " }{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 261 10 "dsolve( \{ " }{TEXT -1 0 "" }{TEXT 262 45 "a differentia
l equation, initial condition(s)" }{TEXT -1 0 "" }{TEXT 263 1 " " }
{TEXT -1 0 "" }{TEXT 264 4 "\},  " }{TEXT -1 22 "the dependent variabl
e" }{TEXT 259 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 31 "I start most sections with the " }{TEXT 274 7 "restart
" }{TEXT -1 55 " command to clear all previous variable definitions by
 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Example 1" }{TEXT 
-1 71 ": Find the general solution to the first order differential equ
ation:  " }{TEXT 283 8 "y' = 3 y" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "sol1 := dsolve(diff(y(x),x) = 3 * y(x), y(x));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%sol1G/-%\"yG6#%\"xG*&%$_C1G\"\"\"-%$expG6#,$F)
\"\"$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Notice: The " }
{XPPEDIT 18 0 "_C1;" "6#%$_C1G" }{TEXT -1 34 " indicates an arbitrary \+
constant. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 9 "Example 2" }{TEXT -1 39 ":   Consider the initial value problem
 " }{TEXT 275 8 "y' = 3 y" }{TEXT -1 5 " ,   " }{TEXT 276 12 "y(0) = 1
/2. " }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 277 1 "a" }{TEXT -1 21 "
) Find the solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol
2 := dsolve(\{diff(y(x),x) = 3*y(x), y(0) = 1/2\}, y(x));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%sol2G/-%\"yG6#%\"xG,$-%$expG6#,$F)\"\"$#\"\"
\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Notice: Maple automati
cally solves for the arbitrary constant based on the initial condition
. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 278 1 "b" }{TEXT 
-1 41 ") Plot the solution from x = 0 to x = 1. " }}{PARA 0 "" 0 "" 
{TEXT -1 96 "First we want to access the right hand side (rhs) of sol2
 and make it a function of x called y2." }}{PARA 0 "" 0 "" {TEXT -1 
22 "This is done with the " }{TEXT 279 7 "unapply" }{TEXT -1 5 " and \+
" }{TEXT 280 3 "rhs" }{TEXT -1 24 " commands as follows.   " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y2 := unapply(rhs(sol2),x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2GR6#%\"xG6\"6$%)operatorG%&ar
rowGF(,$-%$expG6#,$9$\"\"$#\"\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 49 "You can plot the solution from x = 0 to x = 1 by " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(y2(x),x=0..1,labels=[\"
x\",\"y2(x)\"],title=\"The solution to y' = 3y, y(0) = 1/2\");" }}
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)F+$\"1Z5;I!\\[s&Fco7$$\"1nmmm*RRL)F+$\"1ezEFcN#4'Fco7$$\"1mm;a<.Y&)F+
$\"1Y0(zJ&f#\\'Fco7$$\"1LLe9tOc()F+$\"1,Zr.W[:pFco7$$\"1+++]Qk\\*)F+$
\"1'f+Ul<$GtFco7$$\"1LL$3dg6<*F+$\"1UvXEv'=$yFco7$$\"1mmmmxGp$*F+$\"1,
G5;@\\6$)Fco7$$\"1++D\"oK0e*F+$\"1*=xhxo_&))Fco7$$\"1++v=5s#y*F+$\"1T:
BCI-4%*Fco7$$\"\"\"F($\"1%Qfh%oF/5!#9-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-
%+AXESLABELSG6$Q\"x6\"Q&y2(x)Fd[l-%&TITLEG6#QDThe~solution~to~y'~=~3y,
~y(0)~=~1/2Fd[l-%%VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "
(" }{TEXT 281 1 "c" }{TEXT -1 23 ") Evaluate y2 at x = 1 " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(x=1,y2(x));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$-%$expG6#\"\"$#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 53 "and get a numerical value to 6 significant digits by \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(%,6);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"'G/5!\"%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 80 "The % symbol references Maple's last output.  Not necessa
rily the line above it." }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 282 
2 "d)" }{TEXT -1 74 " Find the value of x for which y2 is 100.  Ie. So
lve  y2(x) = 100  for x. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "solve(y2(x) = 100, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG
6#\"$+##\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eval
f(%,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'5m<!\"&" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 7 "Example" }{TEXT -1 
1 " " }{TEXT 268 1 "3" }{TEXT -1 50 ": Plot the solution to the differ
ential equation: " }{TEXT 284 14 "t y' + 2 y = 2" }{XPPEDIT 18 0 "t^2;
" "6#*$%\"tG\"\"#" }{TEXT -1 2 "  " }{TEXT 290 9 "y(1) = 2." }}{PARA 
0 "" 0 "" {TEXT -1 68 "To save on typing we can start by defining the \+
differential equation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode
3 := t*diff(y(t),t)+2*y(t)=2*t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%ode3G/,&*&%\"tG\"\"\"-%%diffG6$-%\"yG6#F(F(F)F)F-\"\"#,$*$)F(F0\"\"
\"F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "and then solve it using t
he " }{TEXT 273 6 "dsolve" }{TEXT -1 8 " command" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "sol3 := dsolve(\{ode3,y(1)=2\}, y(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol3G/-%\"yG6#%\"tG,&*$)F)\"\"#\"\"
\"#\"\"\"F-*&F.F.*$)F)\"\"#F.!\"\"#\"\"$F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "y3 := unapply(rhs(sol3),t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y3GR6#%\"tG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"
\"\"#\"\"\"F0*&F1F1*$)F/\"\"#F1!\"\"#\"\"$F0F(F(F(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 85 "plot(y3(t),t=.5..3,labels=[\"t\",\"y3(t)\"]
,title=\"A plot of the solution to Example 3\");" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7Z7$$\"1+++++++]!#;$
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\"1LL$eMsw[\"F-$\"11mvmcM%y\"F-7$$\"1+DJ&H\"fT:F-$\"1)H!fz6V>=F-7$$\"1
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[.-dp\"F-$\"1(fL#o#o$f>F-7$$\"1n;/Egw[<F-$\"1g0<%py&>?F-7$$\"1n\"z%*f%
)Q!=F-$\"1FwU')4(z3#F-7$$\"1+voza'=&=F-$\"1K'>)Rk4_@F-7$$\"1n;zWho.>F-
$\"1%QWBO:fA#F-7$$\"1++]i>Ad>F-$\"1(ycjUIpI#F-7$$\"1+]i:jf4?F-$\"1Tm#H
zl1R#F-7$$\"1+DJ&>r-1#F-$\"1N%>^6RdZ#F-7$$\"1+]P4q`;@F-$\"1hU5YhquDF-7
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#)>!eXTI#QF-7$$\"1++]i4TPFF-$\"1&R1A/&)o%RF-7$$\"1M$3F9!z#z#F-$\"1\\u9
_S:#4%F-7$$\"1nmmT>KUGF-$\"1-<v'Go]A%F-7$$\"1+DJqJ8&*GF-$\"1)='>%Hd)pV
F-7$$\"1+voa-oXHF-$\"1dl`[iQ6XF-7$$\"\"$\"\"!$\"1nmmmmmmYF--%'COLOURG6
&%$RGBG$\"#5!\"\"Fi\\lFi\\l-%&TITLEG6#QDA~plot~of~the~solution~to~Exam
ple~36\"-%+AXESLABELSG6$Q\"tFg]lQ&y3(t)Fg]l-%%VIEWG6$;$\"\"&Fb]lFg\\l%
(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Again, " }{XPPEDIT 18 0 "_C1;" "6#%
$_C1G" }{TEXT -1 34 " indicates an arbitrary constant. " }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 9 "Example 4" }{TEXT 
-1 64 ": Find and plot the solution to the initial value problem  y' =
 " }{XPPEDIT 18 0 "(3*x^2+4*x+2)/(2*(y-1));" "6#*&,(*&\"\"$\"\"\"*$%\"
xG\"\"#F'F'*&\"\"%F'F)F'F'\"\"#F'F'*&\"\"#F',&%\"yGF'\"\"\"!\"\"F'F3" 
}{TEXT -1 11 ", y(0)=-1. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 
"ode4 := diff(y(x),x) = (3*x^2 + 4*x + 2)/(2*(y(x)-1));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%ode4G/-%%diffG6$-%\"yG6#%\"xGF,*&,(*$)F,\"\"#
\"\"\"\"\"$F,\"\"%F1\"\"\"F2,&F)F1!\"#F5!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "dsolve(\{ode4,y(0)=-1\}, y(x));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&\"\"\"F)*$-%%sqrtG6#,*\"\"%F)*$)F'
\"\"$\"\"\"F)*$)F'\"\"#F3F6F'F6F3!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 82 "Again,  Maple solves for the arbitrary constants based on
 the initial conditions. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Define y4 as
 the right hand side of the previous output" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "y4 := unapply(rhs(%),x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y4GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*$-%%
sqrtG6#,*\"\"%F-*$)9$\"\"$\"\"\"F-*$)F6\"\"#F8F;F6F;F8!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot(y4(x),x=-3..1,labels=[
\"x\",\"y4\"],title=\"A plot of the solution to Example 4\");" }}
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{MPLTEXT 1 0 16 "# end of section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 55 "2. Plotting Direction Fields: The \"dfieldplot\" command." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 291 11 "Example 1: " }{TEXT -1 0 
"" }{TEXT 292 7 "y' = 3y" }}{PARA 0 "" 0 "" {TEXT -1 62 "First we defi
ne the differential equation and give it a name. " }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 31 "ode1 := diff(y(t),t) = 3*y(t); " }}{PARA 11 "
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The command for plotting direction
 fields is in a library called " }{TEXT 257 7 "DEtools" }{TEXT -1 86 "
 in order to access this program we must first load this library. To d
o this, use the " }{TEXT 258 4 "with" }{TEXT -1 9 " command " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "The command to plot direction fields is \+
" }{TEXT 259 10 "dfieldplot" }{TEXT -1 15 " with the usage" }}{PARA 0 
"" 0 "" {TEXT -1 1 " " }{TEXT 256 10 "dfieldplot" }{TEXT -1 81 " (diff
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 256 11 "Example 2: " }{TEXT -1 67 "Have Maple draw the direct
ion field for the differential equation: " }{TEXT -1 6 " y' = " }
{XPPEDIT 18 0 "x^2/(1-y^2);" "6#*&%\"xG\"\"#,&\"\"\"\"\"\"*$%\"yG\"\"#
!\"\"F," }{TEXT -1 4 "   ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "ode2 := diff(y(x),x) = x^2/(1-(y(x))^2);" }}{PARA 11 "" 1 "" 
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0 {PARA 3 "" 0 "" {TEXT -1 10 "Assignment" }}{PARA 0 "" 0 "" {TEXT -1 
420 "You will work on the group problem (Polluted Lake Problem) which \+
is similar to number 20 from section 2.3 in Boyce and Diprima.  You mu
st work in a group of 2-4 members and use the lake denoted on your cov
er sheet.  You have to use Maple to generate the graphs requested in p
arts d and f.  You may use Maple to check your answers on the other pa
rts but I want to see how the answers were derived for parts a,b,c, an
d e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 285 53 "- Five of the points will be based on presentation.  \+
" }}{PARA 0 "" 0 "" {TEXT 286 105 "- You can print Maple graphs right \+
from Maple or export the graphs into another document and print that. \+
" }}{PARA 0 "" 0 "" {TEXT 287 102 "- I only want to see the graphs ...
. NOT all of the commands that went into generating the graph!!!   " }
}{PARA 0 "" 0 "" {TEXT 288 80 "- To enter the text editor in Maple, cl
ick on the \"T\" button from the tool bar. " }}{PARA 0 "" 0 "" {TEXT 
289 72 "- To get a Maple prompt, click on the \"[>\" button from the t
ool bar.    " }{TEXT -1 0 "" }}}}{MARK "3 1 0" 317 }{VIEWOPTS 1 1 0 1 
1 1803 }