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{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 200 22 "Differential Equations" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }} {PARA 0 "" 0 "" {TEXT 268 66 "In this worksheet we demonstrate basic t echniques of working with " }{TEXT 203 31 "ordinary differential equat ions" }{TEXT 268 139 ", (ODE's), using Maple. An ODE is a differential equation in which the unknown function is a function of one variable. We shall stick with " }{TEXT 204 11 "first order" }{TEXT 268 123 " OD E's which contain only the first derivative of the unknown function. M oreover, we shall consider only first order ODE's " }{TEXT 205 18 "in \+ the normal form" }{TEXT 268 22 ", that is in the form:" }}{PARA 206 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = f(x, y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F %F&%\"xGF&!\"\"-%\"fG6$F)F'" }{TEXT 200 2 " ," }}{PARA 0 "" 0 "" {TEXT 268 75 "where the left hand side is simply the derivative of the unkn own function, " }{XPPEDIT 18 0 "y = y(x);" "6#/%\"yG-F$6#%\"xG" }{TEXT 268 137 " in this case, and the right hand side is a given function \+ of x and y. You know that for such ODE's, under reasonable assumptions about " }{XPPEDIT 18 0 "f(x, y);" "6#-%\"fG6$%\"xG%\"yG" }{TEXT 268 40 ", there exists a unique solution to any " }{TEXT 206 23 "initial v alue problem, " }{TEXT 268 21 "( IVP ), of the form " }}{PARA 207 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = f(x, y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F &%\"xGF&!\"\"-%\"fG6$F)F'" }{TEXT 200 5 " , " }{XPPEDIT 18 0 "y(x[0] ) = y[0];" "6#/-%\"yG6#&%\"xG6#\"\"!&F%F)" }{TEXT 200 2 " ," }}{PARA 0 "" 0 "" {TEXT 268 6 "where " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 268 3 " , " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT 268 12 " are given. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 4 "" 0 "" {TEXT 269 26 "Solving and Plotting ODE's" }} {PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 75 "Basic t ools for solving and plotting ODE's are contained in the packages \"" }{TEXT 207 5 "plots" }{TEXT 268 9 "\" and \"" }{TEXT 208 7 "DEtools" } {TEXT 268 98 "\". We begin with loading these packages. Please, don't \+ forget to click on the two commands below." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 35 "We are familiar with the package \"" }{TEXT 209 5 "plots" } {TEXT 268 46 "\". If you are curious about the content of \"" }{TEXT 210 7 "DEtools" }{TEXT 268 87 "\", replace the colon at the end of the command with a semicolon and click on it again." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 211 11 "Example 1. " }{TEXT 268 44 " (a) Find the general solution to the ODE: " }{XPPEDIT 18 0 " d*y/(d*x) = -2*x*y;" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\",$*(\" \"#F&F)F&F'F&F*" }{TEXT 268 2 " ." }}{PARA 0 "" 0 "" {TEXT 268 72 " \+ (b) Solve the following two initial value problems:" }}{PARA 208 "" 0 "" {TEXT 200 5 " " }{XPPEDIT 18 0 "d*y/(d*x) = - 2*x*y;" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\",$*(\"\"#F&F)F&F'F&F *" }{TEXT 200 5 " , " }{XPPEDIT 18 0 "y(0) = 2;" "6#/-%\"yG6#\"\"!\" \"#" }{TEXT 200 22 " , " }}{PARA 0 "" 0 "" {TEXT 268 30 " and" }}{PARA 209 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = -2*x*y;" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\", $*(\"\"#F&F)F&F'F&F*" }{TEXT 200 4 " , " }{XPPEDIT 18 0 "y(0) = 1/2;" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }{TEXT 200 18 " . \+ " }}{PARA 210 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 268 111 " (c) Plot the solutions to the IVP's together \+ with the slope field corresponding to the ODE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 67 "In ord er to simplify many commands below let's first label our ODE:" }} {PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ODE1:=diff(y(x),x)=-2*x*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%ODE1G/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"\"#\"\"\"F,F0F)F0!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 67 "Observe that correct syntax. The derivative is entered u sing the \"" }{TEXT 212 4 "diff" }{TEXT 268 26 "\" command. The comman d \"" }{TEXT 213 7 "D(y)(x)" }{TEXT 268 38 "\" could be used as well. \+ Note that \"" }{TEXT 214 1 "y" }{TEXT 268 26 "\" has to be entered as \+ \"" }{TEXT 215 4 "y(x)" }{TEXT 268 44 "\". The main command for solvin g ODE's is \"" }{TEXT 216 6 "dsolve" }{TEXT 268 4 "\". " }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolv e(ODE1,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1 G\"\"\"-%$expG6#,$*$)F'\"\"#F*!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 39 "Maple return ed the general solution. \"" }{TEXT 217 3 "_C1" }{TEXT 268 178 "\" den otes, of course, an arbitrary constant. Instead of using the name \"OD E1\" you could have entered the differential equation \"diff(y(x),x)=- 2*x*y(x)\" directly into the \"" }{TEXT 218 6 "dsolve" }{TEXT 268 99 "\" command. Maple can handle initial value problems, as well. The pr oper syntax looks as follows. " }}{PARA 0 "" 0 "" {TEXT 268 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "dsolve(\{ODE1,y(0)=2\},y(x)) ; dsolve(\{ODE1,y(0)=1/2\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG,$*&\"\"#\"\"\"-%$expG6#,$*$)F'F*F+!\"\"F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&#\"\"\"\"\"#F+-%$expG6#,$*$)F 'F,F+!\"\"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 111 "Maple can plot slope fields, as \+ well as slope fields together with particular solutions. Proper comman ds are \"" }{TEXT 219 10 "dfieldplot" }{TEXT 268 7 "\" and " }{TEXT 220 8 "\"DEplot" }{TEXT 268 28 "\", both contained in the \"" }{TEXT 221 7 "DEtools" }{TEXT 268 65 "\" package. Let's see how they work. Pa y attention to the syntax." }}{PARA 0 "" 0 "" {TEXT 268 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "dfieldplot(ODE1,[y(x)],x=-2. .2,y=-2..2,color=blue,scaling=constrained,arrows=LINE,dirgrid=[30,30]) ;" }}{PARA 13 "" 1 "" {TEXT 270 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 76 "Maple plotte d the slope field for our equation. All the options under the \"" } {TEXT 222 10 "dfieldplot" }{TEXT 268 152 "\" command regarding color, \+ appearance of the arrows, scaling and dirgrid are, of course, optional . You can play with them and see what will happen. \"" }{TEXT 223 7 " dirgrid" }{TEXT 268 196 "\" tells Maple how dense you want the field o f slopes to be. The default setting is dirgrid[20,20] and it tends to \+ be a little rough. On the other hand, a finer grid may take more time \+ to compute." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 224 170 "Remark. Whenever plotting field of slopes, you should use the \"scaling = constrained\" option. Otherwise, the pictures may appear \+ misleading, as slopes become distorted." }{TEXT 268 1 " " }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 14 "The command \"" } {TEXT 225 6 "DEplot" }{TEXT 268 60 "\" plots the slope field together \+ with particular solutions." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "DEplot(ODE1,y(x),-2..2,[[y( 0)=2],[y(0)=1/2]], linecolor=magenta,color=blue,scaling=constrained,ar rows=LINE);" }}{PARA 13 "" 1 "" {TEXT 270 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 159 "As we \+ expected, the two particular solutions are bell-shaped curves. If you \+ do not want the slope field plotted with particular solutions, you add an option \"" }{TEXT 226 11 "arrows=NONE" }{TEXT 268 15 "\" under the \"" }{TEXT 227 6 "DEplot" }{TEXT 268 11 "\" command." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 228 11 "Example 2. " }{TEXT 268 51 " (a) \+ Try to find the general solution to the ODE: " }{XPPEDIT 18 0 "d*y/(d *x) = cos(x*y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\"-%$cosG6#*& F)F&F'F&" }{TEXT 268 2 " ." }}{PARA 0 "" 0 "" {TEXT 268 40 " \+ (b) Solve the IVP: " }}{PARA 211 "" 0 "" {XPPEDIT 18 0 "d*y /(d*x) = cos(x*y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\"-%$cosG6 #*&F)F&F'F&" }{TEXT 200 6 " , " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\" yG6#\"\"!F'" }{TEXT 200 3 " ." }}{PARA 0 "" 0 "" {TEXT 268 1 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 37 "Let's see how Maple handles this \+ ODE." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ODE2:=diff(y(x),x)=cos(x*y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ODE2G/-%%diffG6$-%\"yG6#%\"xGF,-%$cosG6#*&F,\"\"\"F) F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ODE2,y(x));" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 219 "Maple returned no output! That means Maple is unable \+ to solve the equation. If you are curious what steps Maple went throug h to find a solution before failing to do so, you can ask to see the s teps using the command \"" }{TEXT 229 9 "infolevel" }{TEXT 268 128 "\" . The levels of information that you can request range from 0 to 5. Th e higher number, the more information Maple will reveal." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "info level[dsolve]:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve (ODE2,y(x));" }}{PARA 212 "" 1 "" {TEXT 200 29 "Methods for first orde r ODEs:" }}{PARA 213 "" 1 "" {TEXT 200 38 "trying to isolate the deriv ative dy/dx" }}{PARA 214 "" 1 "" {TEXT 200 29 "successful isolation of dy/dx" }}{PARA 215 "" 1 "" {TEXT 200 33 " -> trying classification me thods" }}{PARA 216 "" 1 "" {TEXT 200 19 "trying a quadrature" }}{PARA 217 "" 1 "" {TEXT 200 23 "trying 1st order linear" }}{PARA 218 "" 1 "" {TEXT 200 16 "trying Bernoulli" }}{PARA 219 "" 1 "" {TEXT 200 16 "try ing separable" }}{PARA 220 "" 1 "" {TEXT 200 21 "trying inverse linear " }}{PARA 221 "" 1 "" {TEXT 200 47 "trying simple symmetries for impli cit equations" }}{PARA 222 "" 1 "" {TEXT 200 25 "trying homogeneous ty pes:" }}{PARA 223 "" 1 "" {TEXT 200 12 "trying Chini" }}{PARA 224 "" 1 "" {TEXT 200 12 "trying exact" }}{PARA 225 "" 1 "" {TEXT 200 44 " -> \+ trying 2nd set of classification methods" }}{PARA 226 "" 1 "" {TEXT 200 14 "trying Riccati" }}{PARA 227 "" 1 "" {TEXT 200 11 "trying Abel" }}{PARA 228 "" 1 "" {TEXT 200 31 " -> trying Lie symmetry methods" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 207 "As you see, Maple tried to match the equation with the \+ types of first order ODE's that it knows how to solve in a closed form , and failed. Out of curiosity, let's see how Maple solved the previou s equation, " }{TEXT 230 4 "ODE1" }{TEXT 268 32 ", with which it was \+ successful." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(ODE1,y(x));" }}{PARA 229 "" 1 "" {TEXT 200 29 "Methods for first order ODEs:" }}{PARA 230 "" 1 "" {TEXT 200 38 "trying to isolate the derivative dy/dx" }}{PARA 231 "" 1 "" {TEXT 200 29 "successful isolation of dy/dx" }}{PARA 232 "" 1 "" {TEXT 200 33 " -> trying classification methods" }}{PARA 233 "" 1 "" {TEXT 200 19 "trying a quadrature" }}{PARA 234 "" 1 "" {TEXT 200 23 "trying 1st \+ order linear" }}{PARA 235 "" 1 "" {TEXT 200 27 "1st order linear succe ssful" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&%$_C1G\"\"\"- %$expG6#,$*$)F'\"\"#F*!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 548 "As you see, Maple solv ed the equation by the first successful method, that is, as a linear e quation in y(x). You don't know that method yet. You could, however, \+ solve the equation by hand as a separable equation. Besides the common type equations listed above, Maple is familiar with more sophisticate d techniques of solving ODE's involving power series expansion, Laplac e transform and others. You have to tell Maple if you want it to apply those techniques. We are not going to do so at this point. Let's get back to the normal infolevel for \"" }{TEXT 231 6 "dsolve" }{TEXT 268 11 "\" command." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "infolevel[dsolve]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 57 "Ma ple couldn't find the general solution of the equation " }{XPPEDIT 18 0 "d*y/(d*x) = cos(x*y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\"-% $cosG6#*&F)F&F'F&" }{TEXT 268 139 " and neither can we. However, the s olution to the IVP of (b), can be found using numerical methods . We \+ describe them in the next section." }}{PARA 0 "" 0 "" {TEXT 268 0 "" } }}{EXCHG {PARA 4 "" 0 "" {TEXT 269 54 "Solving Initial Value Problems \+ Numerically Using Maple" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 217 "Maple is programmed for the whole array of sophis ticated numerical methods for solving ODE's. Let's find the numeric so lution to the IVP of Example 2. (b). The proper syntax for evoking Ma ple's numerical solver is \"" }{TEXT 232 20 "dsolve(....,numeric)" } {TEXT 268 69 "\". You should always label the resulting solution. For \+ example as \"" }{TEXT 233 4 "soln" }{TEXT 268 4 "\". " }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "soln:=d solve(\{ODE2,y(0)=3\},y(x),numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%solnGf*6#%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCop yright~(c)~1993~by~the~University~of~Waterloo.~All~rights~reserved.G6 \"C&>F+-%&evalfGF&@$52-%$absG6#,$F+!\"\"-F96#,&&%,loc_controlG6#\"\"# \"\"\"F+F<4-%'memberG6$&FA6#\"\"'<*$F<\"\"!$!\"#FN$FCFN$FDFNFFA-%%copyG6#=F06#;FD\"#EE\\[l;FDFDFCFN\"\"$FN\"\"&$FD!\")\"\"%Fin\"\" ($FD!\"*FKFD\"#5FN\"#6FN\"\")\"&++$\"\"*\"%+5\"#:FN\"#9FN\"#8FN\"#7FN \"#@FN\"#?$FgnFN\"#BFN\"#AFN\"#;FN\"#FNFenFN\"#CFN\"#DFN> %'loc_y0G-FV6#=F06#;FDFDE\\[l\"FDF[p>%'loc_y1G-FV6#=F0FipE\\[l!@$0F8FN C$>&FA6#FgnF+@%1%'DigitsG-%'evalhfG6#FjqC$>F*-%*traperrorG6#-F\\r6#-%= dsolve/numeric_solnall_rkf45G6,%&loc_FG-%$varG6#FA-Fjr6#Fep-Fjr6#F]q-F jr6#%'loc_F1G-Fjr6#%'loc_F2G-Fjr6#%'loc_F3G-Fjr6#%'loc_F4G-Fjr6#%'loc_ F5G-Fjr6#%)loc_workG@$/F*%*lasterrorGC%>F,-%+searchtextG6$.F\\r-%(conv ertG6$-%#opG6$FD7#F*%%nameG>F--Fht6$.%)hardwareGF[u@%50F,FN0F-FN-Ffr6, FhrFAFepF]qFbsFesFhsF[tF^tFat-%&ERRORG6#F*F\\v7$/%\"xGF'-%$seqG6$/&%$o rdG6#,&F)FDFDFD&Fep6#F)/F)FjpF06%FAFepF]qF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 207 "The outp ut looks like a failure. It is not. Maple reports its numerical soluti on as an algorithm that allows us to calculate values of the solution \+ at any point we want, as well as plot it. The expression \"" }{TEXT 234 5 "rkf45" }{TEXT 268 263 "\" stands for the Fehlberg fourth-fifth \+ order Runge-Kutta method, which is a well-known numerical scheme for s olving ODE's. Maple uses it as a default option. You can guess that th e rkf45 algorithm is much more advanced that the Euler method that we \+ have learned." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 47 "We can find values of our numerical solution \"" }{TEXT 235 4 "soln" }{TEXT 268 82 "\" at any single point we want, or at a li st of points using the following syntax:" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "[soln(1),soln(1.5),s oln(2),soln(2.5),soln(3),soln(3.5),soln(4),soln(4.5),soln(5)];" }} {PARA 236 "" 1 "" {XPPMATH 20 "6#7+7$/%\"xG\"\"\"/-%\"yG6#F&$\"1`KO7=Z sH!#:7$/F&$\"#:!\"\"/F)$\"1A*3:b9**\\#F.7$/F&\"\"#/F)$\"1HyE?y(R?#F.7$ /F&$\"#DF3/F)$\"1DI[h,$eK#F.7$/F&\"\"$/F)$\"1zpf!>muj#F.7$/F&$\"#NF3/F )$\"1gH`am4GCF.7$/F&\"\"%/F)$\"1l5Z^NF@@F.7$/F&$\"#XF3/F)$\"1[m[$)Qsa= F.7$/F&\"\"&/F)$\"1i![>kUgk\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 165 "Maple displays the c onsecutive values of x together with the corresponding values of the s olution y(x). We can also plot a numerical solution to an IVP using t he \"" }{TEXT 236 7 "odeplot" }{TEXT 268 47 "\" command. This command \+ is contained in the \"" }{TEXT 237 5 "plots" }{TEXT 268 97 "\" package , which we have already loaded. The command can be used only with num erical solutions." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "odeplot(soln,[x,y(x)],0..5,color=magenta,th ickness=2,scaling=constrained);" }}{PARA 13 "" 1 "" {TEXT 270 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 100 "It may be interesting to see how this numerical solutio n relates to the slope field of the equation " }{XPPEDIT 18 0 "d*y/(d* x) = cos(x*y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\"-%$cosG6#*&F )F&F'F&" }{TEXT 268 43 ". To see both together you could use the \"" } {TEXT 238 6 "DEplot" }{TEXT 268 36 "\" command. You can also use the \+ \"" }{TEXT 239 7 "display" }{TEXT 268 22 "\" command from the \"" } {TEXT 240 5 "plots" }{TEXT 268 11 "\" package." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "p1:=odeplo t(soln,[x,y(x)],0..5,color=magenta,thickness=2,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "p2:=dfieldplot(ODE2,[y(x )],x=0..5,y=0..5,color=blue,arrows=LINE,scaling=constrained,dirgrid=[3 0,30]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display([p1,p2]) ;" }}{PARA 13 "" 1 "" {TEXT 270 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 269 36 "An Applied E xample: Mr. Jones' Date" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 " " 0 "" {TEXT 241 11 "Example 3. " }{TEXT 268 249 " Mr. Jones has invi ted his date for dinner for 6 p.m. He plans to serve baked chicken. At 5:30 p.m. he realizes that he hasn't started baking the chicken. In a panic, Mr. Jones forgets to preheat the oven. He puts cold chicken fr om the fridge, at " }{XPPEDIT 18 0 "40^0;" "6#*$)\"#S\"\"!\"\"\"" } {TEXT 268 107 "F, into a cold oven and turns the oven on. The tempera ture of the oven is rising according to the function" }}{PARA 237 "" 0 "" {XPPEDIT 18 0 "O(t) = 350-280*exp(-Float(7, -1)*t);" "6#/-%\"OG6#% \"tG,&\"$]$\"\"\"*&\"$!GF*-%$expG6#,$*&-%&FloatG6$\"\"(!\"\"F*F'F*F6F* F6" }{TEXT 200 2 " ," }}{PARA 0 "" 0 "" {TEXT 268 103 "where t is the \+ time, in minutes, after the oven is turned on. The internal temperatur e of the chicken, " }{XPPEDIT 18 0 "H(t);" "6#-%\"HG6#%\"tG" }{TEXT 268 109 ", in degrees Fahrenheit, t minutes after the oven is turned o n, changes according to Newton's Law of Cooling:" }}{PARA 238 "" 0 "" {XPPEDIT 18 0 "d*H/(d*t) = -Float(7, -3)*(H(t)-O(t));" "6#/*(%\"dG\"\" \"%\"HGF&*&F%F&%\"tGF&!\"\",$*&-%&FloatG6$\"\"(!\"$F&,&-F'6#F)F&-%\"OG F4F*F&F*" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 268 58 "The chicke n is done when its internal temperature reaches " }{XPPEDIT 18 0 "175^ 0;" "6#*$)\"$v\"\"\"!\"\"\"" }{TEXT 268 59 "F. At what time will Mr. J ones be able to serve the dinner?" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 21 "To find the function " }{XPPEDIT 18 0 "H(t);" "6#-%\"HG6 #%\"tG" }{TEXT 268 38 ", we have to solve the following IVP:" }} {PARA 239 "" 0 "" {XPPEDIT 18 0 "d*H/(d*t) = -Float(7, -3)*(H(t)-350+2 80*exp(-Float(7, -1)*t));" "6#/*(%\"dG\"\"\"%\"HGF&*&F%F&%\"tGF&!\"\", $*&-%&FloatG6$\"\"(!\"$F&,(-F'6#F)F&\"$]$F**&\"$!GF&-%$expG6#,$*&-F.6$ F0F*F&F)F&F*F&F&F&F*" }{TEXT 200 5 " , " }{XPPEDIT 18 0 "H(0) = 40;" "6#/-%\"HG6#\"\"!\"#S" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 268 27 "Then we can determine when " }{XPPEDIT 18 0 "H(t) = 175;" "6#/-%\" HG6#%\"tG\"$v\"" }{TEXT 268 350 ". Observe that this equation is diffe rent from the other ones we have seen in connection with Newton's Law of Cooling because in our present example we are not assuming that t he temperature of the oven is constant. As a result, the differential \+ equation is no longer separable, we will have trouble solving it by ha nd. Let's hope Maple can help us." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ODE3:=diff(H(t),t)=-(7/1000 )*(H(t)-350+280*exp(-(7/10)*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %ODE3G/-%%diffG6$-%\"HG6#%\"tGF,,(*&#\"\"(\"%+5\"\"\"F)F2!\"\"#\"#\\\" #?F2*&#F5\"#DF2-%$expG6#,$*&#F0\"#5F2F,F2F3F2F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 60 "Maple somewhat simplified the equation. Let's solve the IVP." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "dsolve(\{ODE3,H(0)=40\},H(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6#%\"tG,(\"$]$\"\"\"*&#\"$!G\"#**F*-%$expG6#,$* &#\"\"(\"#5F*F'F*!\"\"F*F**&#\"&q4$F.F*-F06#,$*&#F5\"%+5F*F'F*F7F*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 268 93 "We have replaced all decimals by fractions in our ODE, as in earlier releases of Maple the \"" }{TEXT 242 6 "dsolve" }{TEXT 268 58 "\" command does not work if an equation contains decimals." }} {PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 16 "IMPORTA NT NOTE: " }{TEXT 243 339 "It is very important to realize that the ou tput of the \"dsolve\" command may seem like a function, but, as far a s Maple is concerned, H(t) is neither a function nor an expression. H( t) has not been properly defined as either one. In order to further pr ocess a solution to a given ODE, you have to make it into an expressio n or a function." }{TEXT 200 2 " " }}{PARA 0 "" 0 "" {TEXT 268 0 "" } }{PARA 0 "" 0 "" {TEXT 200 46 "To obtain your solution as an expressio n, say " }{TEXT 244 3 "exH" }{TEXT 200 35 ", you can use the following syntax:" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "dsolve(\{ODE3,H(0)=40\},H(t)); exH:=rhs(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"HG6#%\"tG,(\"$]$\"\"\"*&#\"$!G\"# **F*-%$expG6#,$*&#\"\"(\"#5F*F'F*!\"\"F*F**&#\"&q4$F.F*-F06#,$*&#F5\"% +5F*F'F*F7F*F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$exHG,(\"$]$\"\"\" *&#\"$!G\"#**F'-%$expG6#,$*&#\"\"(\"#5F'%\"tGF'!\"\"F'F'*&#\"&q4$F+F'- F-6#,$*&#F2\"%+5F'F4F'F5F'F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 2 "\"" }{TEXT 245 3 "rhs" } {TEXT 268 20 "\" stands for the \"" }{TEXT 246 15 "right hand side" } {TEXT 268 19 "\". Recall, that \"" }{TEXT 247 1 "%" }{TEXT 268 36 "\" \+ stands for the last output. Now, " }{TEXT 248 3 "exH" }{TEXT 268 65 " \+ is an expression in terms of t and you can use commands, like \"" } {TEXT 249 4 "plot" }{TEXT 268 167 "\" and others with it. If you prefe r to have your solution as a function, and you don't want to type long formulas, you have to use the command with a strange name \"" }{TEXT 250 7 "unapply" }{TEXT 268 94 "\" . The command turns an expression i nto a function as follows. We shall turn the expression " }{TEXT 251 4 "exH " }{TEXT 200 13 "in terms of t" }{TEXT 268 17 " into a function " }{TEXT 252 3 "fH " }{TEXT 268 6 "of t. " }}{PARA 0 "" 0 "" {TEXT 268 3 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fH:=unapply(e xH,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fHGf*6#%\"tG6\"6$%)operat orG%&arrowGF(,(\"$]$\"\"\"*&#\"$!G\"#**F.-%$expG6#,$*&#\"\"(\"#5F.F'F. !\"\"F.F.*&#\"&q4$F2F.-F46#,$*&#F9\"%+5F.F'F.F;F.F;F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 1 "" {TEXT 200 222 "The logic behind the name \"unapply\" is that an expression i n terms of t may be thought of as a result of applying a certain funct ion to a given t. Hence, to turn an expression back into a function we have to \"unapply\"." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 74 "Let's plot the function fH to see what is happening \+ to Mr. Jones' chicken." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(fH(t),t=0..90);" }}{PARA 13 "" 1 "" {TEXT 270 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 101 "We see the the temperature of th e chicken will reach 175 degrees somewhere between t = 60 and t = 90." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(fH(t)=175,t,60..9 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*p?\")H)!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 154 "Well, the chicken will be ready about 83 minutes after 5:30 p .m., that is, about 6:50 p.m. Mr. Jones has to hope that his date does n't arrive too hungry! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 269 14 "Euler's Method" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 426 "Maple can be used to illustrate Euler's method that we have learned. One can say that E uler's method is to numerical methods for ODE's what Left Sums are to \+ numerical integration. It is a rudimetary and not very efficient numer ical method, which, however, is easy to understand and very illuminati ng. Maple has Euler's method built in as one of the options, but in or der to see all the steps, we shall program it from scratch." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 253 12 "Example 4. " }{TEXT 268 78 "Construct approximate solutions for x from 0 to 5 to th e initial value problem" }}{PARA 240 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) \+ = cos(x*y);" "6#/*(%\"dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\"-%$cosG6#*&F)F& F'F&" }{TEXT 200 4 " , " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"! \"\"$" }{TEXT 200 1 " " }}{PARA 0 "" 0 "" {TEXT 254 1 " " }{TEXT 268 58 "using Euler's method with the three different step sizes " } {XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT 268 78 " = 0.5, 0.25, 0.125. Compare your solutions with Maple's solution to the IVP. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 355 "To execute Euler's method, we could go three \+ times step-by-step through ten steps. We could tell Maple in one c ommand to do the ten steps for our particular IVP using the so-called \+ loop structure. Instead, we shall program a simple procedure that calc ulates Euler approximations for any given ODE with a right hand side f (x,y), any initial conditions " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\" !" }{TEXT 268 3 " , " }{XPPEDIT 18 0 "y[0];" "6#&%\"yG6#\"\"!" }{TEXT 268 50 " and any number od steps n. The loop structure \"" }{TEXT 255 7 "do...od" }{TEXT 268 422 "\" is a part of the procedure. This is a simple example of Maple programming. We will program a procedure ca lled \"Eulermethod\" which, given f(x,y), initial conditions (x0,y0), \+ step size h, and number of steps n, will calculate the steps of Euler' s method and display the resulting list of pairs [ [x[0],y[0]], [x[1] ,y[1]], ...,[x[n],y[n] ]. To clarify the structure of our little prog ram we will comment on each step. " }}{PARA 0 "" 0 "" {TEXT 268 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Eulermethod:=proc(f,x0,y0 ,h,n) local i,x,y; \n" }{MPLTEXT 1 0 25 " x[0]:=x0; y[0]:=y0; " } {TEXT 268 59 " first pair of the list is given by the initial condit ion" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 22 " for i from 1 to n " } {TEXT 268 33 "we tell Maple to perform n steps" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 7 " do" }{TEXT 268 37 " mark beginning of the loop \+ commands" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 47 " y[i]:=y[i-1]+ evalf(f(x[i-1],y[i-1]))*h;" }{TEXT 268 21 " get the next y value" } {MPLTEXT 1 0 3 " \n" }{MPLTEXT 1 0 24 " x[i]:=x[i-1]+h; " } {TEXT 268 20 "get the next x value" }{MPLTEXT 1 0 3 " \n" }{MPLTEXT 1 0 9 " od; " }{TEXT 268 33 "mark the end of the loop commands" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 34 " [seq([x[i-1],y[i-1]],i=1..n+1) ];" }{TEXT 268 29 " form the final list of pairs" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 42 "Let's apply the procedure to o ur function " }{XPPEDIT 18 0 "f(x, y) = cos(x*y);" "6#/-%\"fG6$%\"xG% \"yG-%$cosG6#*&F'\"\"\"F(F-" }{TEXT 268 1 "." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->cos(x* y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)oper atorG%&arrowGF)-%$cosG6#*&F(\"\"\"F'F1F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 10 "Lat's us e " }{TEXT 256 12 "Eulermethod " }{TEXT 268 181 "with step sizes h = 0 .5 =1/2 , h = 0.25=1/4, h=0.125=1/8 and the corresponding values of \+ n=10,20,40 to cover the interval [0,5]. We label the resulting list of values by A1,A2,A3." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A1:=Eulermethod(f,0,3,0.5,10);" }} {PARA 241 "" 1 "" {XPPMATH 20 "6#>%#A1G7-7$\"\"!\"\"$7$$\"\"&!\"\"$\"# NF,7$$\"#5F,$\"+sp(3T$!\"*7$$\"#:F,$\"++i*)GHF47$$\"#?F,$\"+.k1sFF47$$ \"#DF,$\"+g*>;9$F47$$\"#IF,$\"+FeeTJF47$F-$\"+FeeTEF47$$\"#SF,$\"+$) \\f\\@F47$$\"#XF,$\"+\"HI3\"=F47$$\"#]F,$\"+Mxdl;F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A2:=Eulermethod(f,0,3,0.25,20);" }}{PARA 242 "" 1 "" {XPPMATH 20 "6#>%#A2G777$\"\"!\"\"$7$$\"#D!\"#$\"$D$F,7$$ \"#]F,$\"+\"R@>U$!\"*7$$\"#vF,$\"+B\\*pQ$F47$$\"$+\"F,$\"+u7&3=$F47$$ \"$D\"F,$\"+/R/JHF47$$\"$]\"F,$\"+VWO9FF47$$\"$v\"F,$\"+Jn*[c#F47$$\"$ +#F,$\"+;yS4DF47$$\"$D#F,$\"+N7#[e#F47$$\"$]#F,$\"+mR,3GF47$$\"$v#F,$ \"+d7;$*HF47$$\"$+$F,$\"+?)y5!HF47$F-$\"+@F47$$\"$ ]%F,$\"+!oPl%=F47$$\"$v%F,$\"+pPdO " 0 "" {MPLTEXT 1 0 32 "A3:=Eulermethod(f,0,3,0.125,40);" } }{PARA 243 "" 1 "" {XPPMATH 20 "6#>%#A3G7K7$\"\"!\"\"$7$$\"$D\"!\"$$\" %DJF,7$$\"$]#F,$\"+3ReSK!\"*7$$\"$v$F,$\"+&*zvELF47$$\"$+&F,$\"+lcYmLF 47$$\"$D'F,$\"+'pSCN$F47$$\"$](F,$\"+@a%)*G$F47$$\"$v)F,$\"+k_>#>$F47$ $\"%+5F,$\"+whquIF47$$\"%D6F,$\"+%o&)*\\HF47$$\"%]7F,$\"+]<%p#GF47$$\" %v8F,$\"+LuU6FF47$$\"%+:F,$\"+%QDtg#F47$$\"%D;F,$\"+eR`#R#F47$$\"%+?F,$\"+j$eSO#F47$$\"%D@F,$\" +]U-mBF47$$\"%]AF,$\"+P.![S#F47$$\"%vBF,$\"+/c<&[#F47$$\"%+DF,$\"+As@, EF47$$\"%DEF,$\"+#G3Ks#F47$$\"%]FF,$\"+,eE/GF47$$\"%vGF,$\"+Gs)>#GF47$ $\"%+IF,$\"+&)\\%**y#F47$F-$\"+@`GGFF47$$\"%]KF,$\"+x]Z]EF47$$\"%vLF,$ \"+YUNkDF47$$\"%+NF,$\"+\"eAYZ#F47$$\"%DOF,$\"+)eGVQ#F47$$\"%]PF,$\"+1 Sg&H#F47$$\"%vQF,$\"+4a)*4AF47$$\"%+SF,$\"+4N`G@F47$$\"%DTF,$\"+a)y=0# F47$$\"%]UF,$\"+7&o-)>F47$$\"%vVF,$\"+#oTO\">F47$$\"%+XF,$\"+Yps^=F47$ $\"%DYF,$\"+\"zQTz\"F47$$\"%]ZF,$\"+QXXSV;F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 192 "Remember, the three lists of pai rs correspond to approximate solutions y(x) on [0,5] provided by the E uler method with smaller and smaller step size. Recall that we conside red the same IVP : " }{XPPEDIT 18 0 "d*y/(d*x) = cos(x*y);" "6#/*(%\" dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\"-%$cosG6#*&F)F&F'F&" }{TEXT 268 3 " , " }{XPPEDIT 18 0 "y(0) = 3;" "6#/-%\"yG6#\"\"!\"\"$" }{TEXT 268 47 " \+ in Sections 1 and 2. The equation was labeled " }{TEXT 257 4 "ODE2" } {TEXT 268 70 " and the numeric solution for the IVP provided by Maple \+ was labeled \"" }{TEXT 258 4 "soln" }{TEXT 268 52 "\". We obtain the p lot of a numeric solution with \"" }{TEXT 259 7 "odeplot" }{TEXT 268 11 "\" command." }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plo1:=odeplot(soln,[x,y(x)],0..5,color=magent a,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " plo2:=pointplot(A1,color=blue,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plo3:=pointplot(A2,color=green,scaling=con strained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plo4:=pointpl ot(A3,color=black,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display([plo1,plo2,plo3,plo4]);" }}{PARA 13 "" 1 "" {TEXT 270 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 83 "As you see, our Euler solutions a re getting closer and closer to Maple's solution. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT 269 17 "Ho mework Problems" }}{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT 268 108 "Do your homework in the same Maple session during which you reexecuted the commands in the above worksheet. " }{TEXT 260 7 "D o not " }{TEXT 268 93 "put the \"restart\" command in your homework wo rksheet. If you do, you will have to reload \"" }{TEXT 261 11 "with(pl ots)" }{TEXT 268 9 "\" and \"" }{TEXT 262 13 "with(DEtools)" }{TEXT 268 58 "\" as well as copy, paste and reexecute the definition of " } {TEXT 263 11 "Eulermethod" }{TEXT 268 13 " procedure. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 11 "Problem 1. " }{TEXT 268 27 " Consider the following ODE" }} {PARA 244 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = y-x^3;" "6#/*(%\"dG\"\"\" %\"yGF&*&F%F&%\"xGF&!\"\",&F'F&*$)F)\"\"$F&F*" }{TEXT 200 2 " ." }} {PARA 0 "" 0 "" {TEXT 268 47 "(a) Find the general solution to the eq uation." }}{PARA 0 "" 0 "" {TEXT 268 84 "(b) Plot the slope field corr esponding to the equation for x and y between -2 and 2." }}{PARA 0 "" 0 "" {TEXT 268 37 "(c) Solve the initial value problems" }}{PARA 245 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = y-x^2;" "6#/*(%\"dG\"\"\"%\"yGF&*&F %F&%\"xGF&!\"\",&F'F&*$)F)\"\"#F&F*" }{TEXT 200 5 " , " }{XPPEDIT 18 0 "y(0) = 1/2;" "6#/-%\"yG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }{TEXT 200 2 " ," }}{PARA 246 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = y-x^2;" "6#/*(%\"dG \"\"\"%\"yGF&*&F%F&%\"xGF&!\"\",&F'F&*$)F)\"\"#F&F*" }{TEXT 200 5 " , \+ " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 268 121 "(d) Plot the two solutions and th e slope field in one coordinate system for x between -2 and 2 using th e DEplot command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 265 11 "Problem 2. " }{TEXT 268 70 " The \+ growth of a certain animal population is governed by the equation" }} {PARA 247 "" 0 "" {XPPEDIT 18 0 "d*P/(d*t) = P(t)/2-P(t)^2/1100;" "6#/ *(%\"dG\"\"\"%\"PGF&*&F%F&%\"tGF&!\"\",&*&-F'6#F)F&\"\"#F*F&*&)F-F/F& \"%+6F*F*" }{TEXT 200 3 " ," }}{PARA 0 "" 0 "" {TEXT 268 6 "where " } {XPPEDIT 18 0 "P(t);" "6#-%\"PG6#%\"tG" }{TEXT 268 99 " denotes the nu mber of animals at time t, t is measured in months. There are 200 anim als initially." }}{PARA 0 "" 0 "" {TEXT 268 11 "(a) Find " } {XPPEDIT 18 0 "P(t);" "6#-%\"PG6#%\"tG" }{TEXT 268 2 ". " }}{PARA 0 "" 0 "" {TEXT 268 11 "(b) Graph " }{XPPEDIT 18 0 "P(t);" "6#-%\"PG6#%\" tG" }{TEXT 268 77 " in a large enough interval to see the longterm beh avior of the population. " }}{PARA 0 "" 0 "" {TEXT 268 48 "(c) When \+ will the population reach 400 animals?" }}{PARA 0 "" 0 "" {TEXT 266 4 "Hint" }{TEXT 268 35 ": Remember to define the solution " }{XPPEDIT 18 0 "P(t);" "6#-%\"PG6#%\"tG" }{TEXT 268 94 " to your ODE as an expre ssion or a function, as in Example 3, before attempting to process it. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 11 "Problem 3. " }{TEXT 268 17 " Consider the IVP" }} {PARA 248 "" 0 "" {XPPEDIT 18 0 "d*y/(d*x) = y/110+exp(-x);" "6#/*(%\" dG\"\"\"%\"yGF&*&F%F&%\"xGF&!\"\",&*&F'F&\"$5\"F*F&-%$expG6#,$F)F*F&" }{TEXT 200 3 " , " }{XPPEDIT 18 0 "y(0) = 0;" "6#/-%\"yG6#\"\"!F'" } {TEXT 200 2 " ." }}{PARA 0 "" 0 "" {TEXT 268 61 "(a) Solve the IVP num erically using Maple's numerical solver." }}{PARA 0 "" 0 "" {TEXT 268 74 "(b) Plot the solution in the interval [0,4] using the \"odeplot\" \+ command." }}{PARA 0 "" 0 "" {TEXT 268 94 "(c) For three different valu es of step size h calculate the corresponding Euler approximation." }} {PARA 0 "" 0 "" {TEXT 268 89 "(d) Display your Euler approximations an d the Maple's solution in one coordinate system. " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 249 "" 0 "" {TEXT 200 75 "M TH 142 Maple Worksheets written by B.Kaskosz and L.Pakula, Copyright 1 999." }}}{EXCHG {PARA 250 "" 0 "" {TEXT 200 27 "Last modified August 1 999. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }