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}0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 267 16 "Direction Fields" }} {PARA 258 "" 0 "" {TEXT 256 4 "HW 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expan d a section. Click on a [-] sign to collapse a section. To do these ex ercises you will have to insert execution groups. That can be done by \+ clicking on the toolbar icon that looks like \"[>\". It can also be do ne via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 271 " " 0 "" {TEXT 331 73 "If you do your work as part of a team, then fill \+ in both lines below and " }{TEXT 333 32 "hand in only one report per t eam" }{TEXT 334 36 ". Otherwise, delete the second line." }{TEXT 332 1 "\n" }}{PARA 0 "" 0 "" {TEXT 329 20 "Student Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 330 20 "Student Name a nd ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " } {TEXT 257 5 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "wor ksheet" "" }{TEXT -1 94 ", you will be asked to plot direction fields of first order differential equations. The main" }{TEXT 280 7 " MAP LE" }{TEXT -1 36 " functions that can be used are " }{TEXT 358 4 " diff" }{TEXT -1 3 ", " }{TEXT 359 6 "DEplot" }{TEXT -1 3 ", " } {TEXT 357 6 "dsolve" }{TEXT -1 4 ", " }{TEXT 281 10 "dfieldplot" } {TEXT -1 48 ". Information on, and examples of, the use of " }{TEXT 258 10 "dfieldplot" }{TEXT -1 45 " may be found in the (read-only) wo rksheet " }{HYPERLNK 17 "dfieldplot.mws" 1 "dfieldplotR4.mws" "" } {TEXT -1 2 ". " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Reports (Read! )\n" }}{PARA 0 "" 0 "" {TEXT -1 206 "There are two main themes to this section: (1) reports should be sensibly prepared, and (2) the resourc es used in preparing a report should be considered. This section is wr itten with regard to the use of " }{TEXT 342 5 "MAPLE" }{TEXT 343 1 " " }{TEXT -1 77 " but if you use a different package, then you should \+ use the same discretion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 335 30 "Reports that you prepare with " }{TEXT 282 5 "MAP LE" }{TEXT 336 112 " should be prepared with the same care that you \+ would devote to laboratory reports in biology and chemistry. " } {TEXT 284 46 "A report should not be a diary or history of a" }{TEXT 337 1 " " }{TEXT 283 1 " " }{TEXT 360 5 "MAPLE" }{TEXT 338 2 " " } {TEXT 285 52 " session. Delete what is not needed for the report." } {TEXT 339 25 " All lines of the form " }{TEXT 268 6 "?topic" }{TEXT 340 80 " (that arise from help queries) should be erased. All errors should be erased." }{TEXT 341 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 286 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper resources that you are using. All commands must be terminat ed - either with the standard terminator, the semicolon, or the " } {TEXT 290 17 "silent terminator" }{TEXT -1 57 ", the colon. When you a ssign a variable, for example \n \"" }{TEXT 324 7 "x := 5;" }{TEXT -1 30 " \", there is no need to have " }{TEXT 287 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 269 6 "x := 5" }{TEXT -1 89 ". When this \+ is printed, it simply wastes paper and ink. Choose the silent termina tor \"" }{TEXT 270 7 "x := 5:" }{TEXT -1 70 " \" instead. When you \+ load a package (without the silent terminator)," }{TEXT 271 1 " " } {TEXT -1 2 " " }{TEXT 288 5 "MAPLE" }{TEXT -1 123 " will list the com mands that become available with the package. This is fine - it will h elp you become familiar with what " }{TEXT 289 5 "MAPLE" }{TEXT -1 125 " makes available. However, these commands should not be part of a lab report. Reload the package with the silent terminator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Much of the tex t in this worksheet should be deleted. For example, delete the " } {TEXT 275 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 276 8 "Keywords " }{TEXT -1 34 " sections. Delete this section on " }{TEXT 277 7 "Repo rts" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 235 "Remember that your worksheet should execute in the ord er that it has been written. In particular, remember that the ditto r efers to the result of the last executed command - not the result of t he command that physically precedes it. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Keywords" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "color" 2 "color" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "DEplot" 2 "DEpl ot" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "DEtools" 2 "DEtools" "" } {TEXT -1 4 ", " }{HYPERLNK 17 "dfieldplot" 2 "dfieldplot" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "diff" 2 "diff" "" }{TEXT -1 4 ", " } {HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 3 ", " } {HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 4 ", " }{HYPERLNK 17 " limit" 2 "limit" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "plot" 2 "plot" " " }{TEXT -1 4 ", " }{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 4 ", \+ " }{HYPERLNK 17 "restart" 2 "restart" "" }{TEXT -1 4 ", " } {HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "testeq" 2 "test eq" "" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Backgro und Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 422 "The following workshee ts, available for download from the syllabus web page, have examples \+ or discussions that will help you do this homework. If they are in th e same directory as this worksheet, and if you have retained the filen ame under which they were posted, then clicking on the hyperlink below will automatically open them. Use the Window menu to control the view when multiple files are opened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{HYPERLNK 17 "1.1 epR4.mws" 1 "1.1epR4.mws" "" }{TEXT -1 8 " , " }{HYPERLNK 17 "dfi eldplotR4.mws" 1 "dfieldplotR4.mws" "" }{TEXT -1 6 " , " } {HYPERLNK 17 "Tutor1R4.mws" 1 "Tutor1R4.mws" "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Exercise 1 Checking that a Differential Equation is Satisfied" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 18 "Backgroud Reading:" }{TEXT -1 21 " The section titled \"" } {TEXT 292 30 "A Simple Differential Equation" }{TEXT -1 17 "\" of work sheet " }{HYPERLNK 17 "1.1epR4.mws" 1 "1.1epR4.mws" "" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 59 " \+ The help pages for " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 5 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verify that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 18 " " }} {PARA 0 "" 0 "" {TEXT -1 13 " " }{TEXT 364 8 " " } {XPPEDIT 365 1 "y(x) = A*exp((15*ln(x)*x^(3/2)-4)/12/x^(3/2)) + B*exp( (15*ln(x)*x^(3/2)+4)/12/x^(3/2))" "/-%\"yG6#%\"xG,&*&%\"AG\"\"\"-%$exp G6#*(,&*(\"#:F*-%#lnG6#F&F*)F&*&\"\"$F*\"\"#!\"\"F*F*\"\"%F9F*\"#7F9)F &*&F7F*F8F9F9F*F**&%\"BGF*-F,6#*(,&*(F1F*-F36#F&F*)F&*&F7F*F8F9F*F*F:F *F*F;F9)F&*&F7F*F8F9F9F*F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 363 7 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "is a solution of the differential equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " \007 " }{TEXT 361 14 " \+ " }{XPPEDIT 362 1 "diff(y(x),x,x) = (4+5*x^3)*y(x)/(16*x^5) " "/-%%diffG6%-%\"yG6#%\"xGF)F)*(,&\"\"%\"\"\"*&\"\"&F-*$F)\"\"$F-F-F- -F'6#F)F-*&\"#;F-*$F)F/F-!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "Do this by replacing the question marks in the next two execution groups with appropriate arguments. Execute each gr oup by placing the cursor within it and pressing the Enter key." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "? := subs( ? , ? );" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "If it is not obvious t hat the two sides are equal then try\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "testeq( ? ); # \"test equality\" Erase if not need ed." }}}{PARA 0 "" 0 "" {TEXT -1 10 "\n\n\nIf " }{TEXT 366 33 "eqn := expression1 = expression2" }{TEXT -1 27 " is a true equation b ut " }{TEXT 373 13 " testeq(eqn) " }{TEXT -1 11 " returns " }{TEXT 372 4 "FAIL" }{TEXT -1 32 " then sometimes simplifying\n " }{TEXT 367 20 "hs(eqn) - rhs(eqn) " }{TEXT -1 65 " ( to see if 0 results \+ ) is more powerful than simplifying " }{TEXT 368 4 " eqn" }{TEXT -1 12 " itself.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " si mplify( ? ); # simplify lhs - rhs (Erase if not needed.)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 45 "Exer cise 2 Trajectories and Direction Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 18 "Backgroud Reading:" }{TEXT -1 58 " Worksheet: for background on direction field plots " } {HYPERLNK 17 "dfieldplotR4.mws" 1 "dfieldplotR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 106 " \+ for background on working with functions " }{HYPERLNK 17 "Tutor1R4.mws" 1 "Tutor1R4.mws" "" }{TEXT -1 93 " \+ \nThe help pages for " }{HYPERLNK 17 "dfieldplot" 2 "DEtools[dfieldplot]" "" }{TEXT -1 8 " and " }{HYPERLNK 17 "display" 2 "plots[display]" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 120 "The height of a species of New Ze aland pine tree grows according to the Chapman-Richards equation\n\n\n \n\n " }{TEXT 344 12 " " }{XPPEDIT 345 1 " diff(H(t),t) = 0.072*H(t)*((0.0167*H(t))^(-0.556)-1)" "/-%%diffG6$-% \"HG6#%\"tGF)*($\"#s!\"$\"\"\"-F'6#F)F.,&)*&$\"$n\"!\"%F.-F'6#F)F.,$$ \"$c&F-!\"\"F.F.Fw=Z:V\\EG:MZ=V]JWZ:j:f::=B:kJ:>:: ::vyyyyy=::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::V:>:1:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "a) Direction Field Plot" }}{PARA 0 "" 0 "" {TEXT -1 71 "Plot the direction field of this differential equation in the wind ow " }{TEXT 278 24 "[ 0 , 150 ] x [ 1 , 60 ]" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 122 "Do this by replacing the question marks \+ with appropriate arguments. For the last execution group, look up the available " }{HYPERLNK 17 "color choices" 2 "color" "" }{TEXT -1 105 " and pick one you like. (Light colors do not alwayys print out \+ well.) Execute each line in succession." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart: with(plots ): with(DEtools): # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "? := diff(H(t),t) = 0.072*H(t)*((0.0167*H(t))^(-0.55 6)-1) ; # Name the differential equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dfieldplot( ? , ? , t = ?, H = ? , color = ? );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "Before continuing, choose a name and assign t he plot structure that you have just created to that name. Use the di tto operator on the right side of the assignment" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := ? :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 " b) Plotting the Solution of One Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The function" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 " \+ " }{TEXT 389 18 " " }{XPPEDIT 390 1 "H(t)=60*(1 - e xp(-.04*t))^(9/5)" "/-%\"HG6#%\"tG*&\"#g\"\"\"),&F)F)-%$expG6#,$*&$\" \"%!\"#F)F&F)!\"\"F4*&\"\"*F)\"\"&F4F)" }}{PARA 0 "" 0 "" {TEXT -1 12 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "is a solution of the Chap man-Richards equation whose direction field has been plotted." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Plot " }{TEXT 349 4 "H(t)" }{TEXT -1 8 " for " }{TEXT 348 1 "t" }{TEXT -1 20 " in the interval " }{TEXT 347 10 "[ 0 , 150]" }{TEXT -1 25 ". \+ Use the plot option " }{TEXT 351 13 "thickness = 2" }{TEXT -1 9 ". \+ (See " }{HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 4 " . )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot(? , ? , thickn ess = 2, color = ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Before continuing, name the plot structure that you \+ just created." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "? := replace_this_string_with_ditto_operator :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 63 "c) Superimposing the solution curve on the direction field plot" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Superimp ose the plots that you obtained in parts " }{TEXT 391 2 "a)" }{TEXT -1 6 " and " }{TEXT 392 2 "b)" }{TEXT -1 77 ". (This will look nices t if you coordinated your colors.) \nUse the command " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display( \{ ? , ? \} );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 260 " " 0 "" {TEXT -1 26 "Exercise 3 Direction Field" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 327 18 "Backgroud Reading:" } {TEXT -1 14 " Worksheet " }{HYPERLNK 17 "1.3epR4.mws" 1 "1.3epR4.mw s" "" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 " \+ The help pages for " }{HYPERLNK 17 "dfieldplo t" 2 "DEtools[dfieldplot]" "" }{TEXT -1 5 " , " }{HYPERLNK 17 "displ ay" 2 "plots[display]" "" }{TEXT -1 13 " , and " }{HYPERLNK 17 " subs" 2 "subs" "" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "First, restart the " }{TEXT 310 5 "MAPLE" } {TEXT -1 22 " kernel and load the " }{TEXT 311 7 "DEtools" }{TEXT -1 68 " package. (Place the cursor anywhere in the next line and press th e " }{TEXT 312 5 "Enter" }{TEXT -1 6 " key.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: with(DE tools):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "? := diff(y(x), x) = sin(x - y(x)); # Name this differential e quation." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Don't forget to execute the preceding execution group (by placing the cursor anywhere within and pressing the " }{TEXT 313 5 "Enter" } {TEXT -1 6 " key.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 51 "a) Using the Dire ction Field Plot to Spot Solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " } {TEXT 315 10 "dfieldplot" }{TEXT -1 76 " to plot the direction field \+ of the given equation in the viewing window " }{TEXT 316 21 "[-10, 1 0] x [-10, 10]" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dfieldplot(?, ?, ?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "You should see strong evidence that there are \+ a dozen straight line solutions that intersect this viewing window." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Use the \+ " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 27 " command to subst itute " }{TEXT 317 14 "y(x) = m*x + b" }{TEXT -1 57 " into the giv en equation. Then simplify the equation (" }{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(?, ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify ( ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "You will see that the left hand side is constant. The right hand s ide can therefore not depend on " }{TEXT 318 1 "x" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 36 "What can you conclude the value of " } {TEXT 319 1 "m" }{TEXT -1 5 " is?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "m := ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "solv e" 2 "solve" "" }{TEXT -1 7 " or " }{HYPERLNK 17 "fsolve" 2 "fsolve " "" }{TEXT -1 80 " (with the three parameters: equation, variable, r ange) to find the value of " }{TEXT 320 1 "b" }{TEXT -1 25 " that i s least negative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve( ? , b ); # Use this ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "fsolv e( ? , b, ? .. ? ); # or this, whichever returns the desired value" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "b ) The General Solution" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 321 5 "MAPLE" }{TEXT -1 59 " to find a general solution to the diffe rential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dsolve(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The solution that " }{TEXT 314 5 " MAPLE" }{TEXT -1 46 " returns might be implicit . If so then use " } {MPLTEXT 1 0 5 "solve" }{TEXT -1 16 " to solve for " }{MPLTEXT 1 0 4 "y(x)" }{TEXT -1 13 " explicitly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "eqn := y(x) = solve(? , ?); # This line may be unnecessary wit h some Maple releases." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "One particular solution of the differential equation is " }{MPLTEXT 1 0 15 "y(x) = x - Pi/2" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 47 "There is no value of the arbitrary constant " } {MPLTEXT 1 0 3 "_C1" }{TEXT -1 75 " that corresponds to this solutio n (or any other straight line solution)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "However, plot the solutions that c orrespond to " }{MPLTEXT 1 0 8 "_C1 = 10" }{TEXT -1 8 " and " } {MPLTEXT 1 0 9 "_C1 = 100" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(subs(_C1 = 10, rhs(eqn)), x = -10 .. 10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "plot(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "These plots should g ive you a clue as to how to recover the particular solution " } {MPLTEXT 1 0 15 "y(x) = x - Pi/2" }{TEXT -1 37 " from the general sol ution given by " }{MPLTEXT 1 0 3 "eqn" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Namely, calculate the " }{HYPERLNK 17 "limit" 2 "lim it" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " " }{TEXT 325 10 " " }{XPPEDIT 326 1 "limit(x-2*arctan((-2+x+_C1)/(x+_C1)),_C1=infinity)" "-%&limitG6 $,&%\"xG\"\"\"*&\"\"#F'-%'arctanG6#*&,(F)!\"\"F&F'%$_C1GF'F',&F&F'F0F' F/F'F//F0%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y(x) = limit(?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Exercise 4 The " }{TEXT 309 5 " MAPLE" }{TEXT -1 12 " Function " }{TEXT 308 6 "DEplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 328 18 "Backgroud Reading: " }{TEXT -1 21 " The help page for " }{HYPERLNK 17 "DEplot" 2 "DEtoo ls[DEplot]" "" }{TEXT -1 105 " is rather involved. The next execution group provides you with an example of how to use the function. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 7 "Caveat:" }{TEXT -1 2 " " }{TEXT 377 30 "Unless a stepsize is specified" } {TEXT 378 1 " " }{TEXT -1 3 " " }{TEXT 369 6 "DEplot" }{TEXT -1 3 " \+ " }{TEXT 371 237 "uses a default number of steps. If you use too gr eat an interval then the resulting step size may be too large. If your plot suggests that this is happening, then go to the linked help page for DEplot to see how to adjust the step size." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Look at the commands in t he next execution group to see how we use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 50 " to plot a solution to the Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "\022 " }{TEXT 374 1 " " }{TEXT 376 5 " " } {XPPEDIT 375 1 "diff(y(x), x) = sin(x/(1+x^4)+y(x)),y(0)=2" "6$/-%%dif fG6$-%\"yG6#%\"xGF*-%$sinG6#,&*&F*\"\"\",&F0F0*$F*\"\"%F0!\"\"F0-F(6#F *F0/-F(6#\"\"!\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 54 "Please read and then execute the next execution group. 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AXESLABELSG6$%\"xG%%y(x)G" 2 901 416 416 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 28 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Click on the plot and you will see the tracker coordinates above \+ the worksheet, to the left of the plot toolbar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "a) Behavior At Infinity" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 61 "Notice that the graph of the solution becomes rather flat as " }{TEXT 298 1 "x" }{TEXT -1 15 " increases t o " }{TEXT 296 3 " 20" }{TEXT -1 8 ". Use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 42 " to study the behavior for values of \+ " }{TEXT 297 1 "x" }{TEXT -1 9 " between " }{MPLTEXT 1 0 3 "150" } {TEXT -1 6 " and " }{MPLTEXT 1 0 3 "200" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "DEplot(? , ? , ?, ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Describe what appea rs to be happening to" }{TEXT 380 5 " y(x)" }{TEXT -1 6 " as " } {TEXT 379 1 "x" }{TEXT -1 20 " tends to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "b) Two Other Ini tial Value Problems" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 49 " to study solutions to the In itial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 8 " " }{TEXT 381 1 " " }{TEXT 383 5 " " } {XPPEDIT 382 1 "diff(y(x), x) = sin(x/(1+x^4)+y(x)),y(0)=-10" "6$/-%%d iffG6$-%\"yG6#%\"xGF*-%$sinG6#,&*&F*\"\"\",&F0F0*$F*\"\"%F0!\"\"F0-F(6 #F*F0/-F(6#\"\"!,$\"#5F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 388 9 " " }{XPPEDIT 387 1 "diff(y(x) , x) = sin(x/(1+x^4)+y(x)),y(0)=6" "6$/-%%diffG6$-%\"yG6#%\"xGF*-%$sin G6#,&*&F*\"\"\",&F0F0*$F*\"\"%F0!\"\"F0-F(6#F*F0/-F(6#\"\"!\"\"'" } {TEXT -1 1 " " }{TEXT 386 5 " " }}{PARA 0 "" 0 "" {TEXT 274 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{TEXT 307 1 "x" }{TEXT -1 11 " between " }{TEXT 303 1 "0" }{TEXT -1 7 " and " }{TEXT 302 2 "20" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "DEplot(? , ? , ?, ? , ? );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "DEplot(? \+ , ? , ?, ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Based on the three examples that you have seen, what would you \+ conjecture are the possible" }}{PARA 0 "" 0 "" {TEXT -1 9 "values of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ " }{TEXT 304 4 " " }{XPPEDIT 305 1 "limit(y(x),x =infinity)" "-%&limitG6$-%\"yG6#%\"xG/F(%)infinityG" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "if " }{TEXT 306 4 "y(x )" }{TEXT -1 191 " satisfies the given equation? (Click on the curv es to guess this limit. For the first you should be able to see what \+ exact number the limit is. Use that value as a hint for the second.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 100 "c) Using a \+ Differential Equation to Understand Solution Curves (without having an explicit formula) " }}{PARA 0 "" 0 "" {TEXT -1 74 " Use the different ial equation to explain the reason for the behavior of " }{TEXT 323 4 "y(x)" }{TEXT -1 6 " as " }{TEXT 322 1 "x" }{TEXT -1 73 " tends to infinity. (You should aim for plausibility rather than rigor.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and A uthor Information" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 45 "01S02R4.mws A MapleV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 43 "Author: Brian E. Blank (11 February 2002)" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, with out" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the au thor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Departmen t of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Uni versity in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, M O 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 200 2 Brian E. Blank, All Rights Reserved." }}}}}{MARK "8 19 25 0 0" 72 } {VIEWOPTS 1 1 0 3 4 1802 }