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273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 20 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 276 1 {CSTYLE "" -1 -1 "" 0 1 20 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 285 45 "Numerical Solutions of Di fferential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 62 "Enter the following commands to load the two package s we need." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (plots):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with (DEtools):" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 257 28 "(Part 1 ne eds no responses.)" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 50 "Part 2. Numerical methods in a helper application" }} {PARA 262 "" 0 "" {TEXT 262 5 " " }}{PARA 263 "" 0 "" {TEXT -1 52 "Enter the following command to define the function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 65 " that describes the right-hand side of th e differential equation" }}{PARA 264 "" 0 "" {TEXT 258 35 " \+ " }{XPPEDIT 18 0 "dy/dt=f(t,y)" "6#/*&%#dyG\" \"\"%#dtG!\"\"-%\"fG6$%\"tG%\"yG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y:='y': f:=(t,y)->2*cos(t)-t*y; DE1:=diff(y(t ),t)=f(t,y(t));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 60 "Construct the direc tion field for the differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "y:='y': dfieldplot(DE1, y(t) , t=-1..8, y=-1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 14 "Euler's Method" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 265 "" 0 "" {TEXT -1 45 "First, we use Euler's Method to sol ve the IVP" }}{PARA 266 "" 0 "" {TEXT -1 19 " " } {XPPEDIT 18 0 "dy/dt=2*cos(t)-t*y" "6#/*&%#dyG\"\"\"%#dtG!\"\",&*&\"\" #F&-%$cosG6#%\"tGF&F&*&F/F&%\"yGF&F(" }{TEXT -1 8 " with " } {XPPEDIT 18 0 "y(0)=2" "6#/-%\"yG6#\"\"!\"\"#" }}{PARA 267 "" 0 "" {TEXT -1 44 "on the interval [0,8]. We set the number " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 30 " of steps and the step size " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 39 ", and we calculate t he t-coordinates " }{XPPEDIT 18 0 "t(k)=k*Delta" "6#/-%\"tG6#%\"kG*&F '\"\"\"%&DeltaGF)" }{TEXT 259 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "n:=20; Delta:=8/n; \+ t:=k->k*Delta;" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 57 "Next we define the recursive procedure for getting each " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 144 " from the preceding point. Then we enter the initial \+ condition, calculate the approximate solution points, and collect the \+ points into a list." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "y:='y':\ny:=proc(k) \n y(k):=y(k-1)+f(t(k-1),y(k- 1))*Delta;\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "y(0):=2:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq([y(k)],k=1..n):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "Data(n):=[seq([t(k),y(k)], k=0..n)]:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 95 " Now we plot the approximation solution, giving it a name that depends \+ on the number of points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Egraf(n):=plot(Data(n), style=point , symbol=circle, color=blue):%;" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 107 "After you have displ ayed the approximations for n = 20, 40, and 80, you may display t hem all together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 " " {TEXT -1 22 "Scroll back to where " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT -1 94 " was set to 20, and change it, first to 40, then to \+ 80. Each time you plot, change the " }{TEXT 256 6 "symbol" }{TEXT 284 1 " " }{TEXT -1 8 "and the " }{TEXT 257 5 "color" }{TEXT -1 305 " \+ so you can tell the plots apart when you display them together. (We s uggest that you not use red as one of the colors -- we will later plot these approximate solutions on a red direction field.) Available sym bols include BOX, CROSS, CIRCLE, POINT, and DIAMOND. Most ordinary co lor names are available." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "display([Egraf(20),Egraf(40),Egraf(80)]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 51 "Use this space to descri be the changes you see as " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 267 26 " goes from 20 to 40 to 80" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "T" }{TEXT 260 26 "he Improved Euler's Method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 142 "The following commands repeat what we d id with Euler's Method, but this time the recursive procedure uses the Improved Euler (or Heun) Method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "n:=20; Delta:=8/n; t:=k->k*Delta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "y:='y':\ny:=proc(k) y(k):=y(k-1) +(1/2)*(f(t(k-1),y(k-1))+f(t(k),y(k-1)+f(t(k-1),y(k-1))*Delta))*Delta; \nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "y(0):=2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq([y(k)],k=1..n):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IEdata(n):=[seq([t(k),y(k)], k=0..n)]:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "IEgraf(n):=plot(IEdata(n), style=point, symbo l=circle, color=blue):%;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 271 "" 0 "" {TEXT -1 41 "As you did before, scroll back to where " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 175 " was set to 20, and change it, first to 40, then to 80. Also change the symbol and th e color in the plot so you can tell the plots apart when you display t hem together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 107 "After you have displayed the approximations for n = 20, 40, and 80, you may dis play them all together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "display([IEgraf(20), IEgraf(40), IEgraf(80)]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 51 "Use this space to d escribe the changes you see as " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 269 26 " goes from 20 to 40 to 80" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 27 "Maple's \+ Built-in IVP Solver" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 142 "Maple uses a Fourth-Order Runge-Kutta Method as the d efault for its IVP solver. The following commands plot this approximat ion with 80 steps. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:='y':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "DE1 : = diff(y(t),t)=2*cos(t)-t*y(t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " RKgraf:=DEplot(DE1, y(t), 0..8, \{[0,2]\}, stepsize=0.1, y=-1..3, line color=black):%;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 " " 0 "" {TEXT -1 150 "Now compare RKgraf with the results from the othe r numerical schemes. Vary the second entry to display each of the pre ceding plots along with RKgraf." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display([RKgraf, Egraf(20)]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT -1 30 "Answer questions 5 and \+ 6 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 30 "Part 3. The Log istic Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 35 "Restart Maple and clear the memory." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 51 "We define the logistic differential equation as DE2 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f:=(t,y)->y*(1-y):\nDE2 := diff(y(t),t)=f(t,y(t));" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 71 "Plot the direction field and Runge-Kutta solution starting at [0,0.1 ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "RKplot := DEplot(DE2, y(t), t=0..8, \{[0,0.1]\}, stepsize=0.1, y=0 ..1.5, linecolor=black):%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 38 "Enter your exact solution formula as " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT 273 76 ". You may use Maple for calculations if necessary. For your convenience, " }{XPPEDIT 18 0 "e" "6#%\"eG" }{TEXT -1 1 " " }{TEXT 274 15 " is defined as " }{TEXT -1 1 " " } {TEXT 275 7 "exp(1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "e:=exp(1): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g := t -> ***;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 58 "T he following commands construct a plot of your function " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT 277 67 " and overlay the functio n plot on the RK plot and direction field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "gplot := plot(g(t), t=0.. 8, y=0..1.5, color=blue, thickness=3): display(\{RKplot, gplot\});" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 120 "Scroll back and change \+ the initial point in DEplot (the fourth entry) to [0,1.5]. Recalcula te your solution function " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT 279 29 " and answer question 3 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 38 "Part 4. Another Initial Value Problem" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 181 "Copy, paste, and edit Maple commands from Part 3 here to carry out the calculations for Par t 4. (You may have to adjust horizontal and vertical scales to see wh at you need to see.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 16 "Part 5. Summary" }} {PARA 276 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 36 "Answer the questions in Part 5 here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 3 0" 13 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }