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2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 279 16 "Direction Fields" }} {PARA 258 "" 0 "" {TEXT 280 9 "Exercises" }}{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 expand a section. Click on a \+ [-] sign to collapse a section. To do these exercises you will have to insert execution groups. That can be done by clicking on the toolbar \+ icon that looks like \"[>\". It can also be done via the Insert menu. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 399 19 "Stud ent Name and ID" }{TEXT 401 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 400 19 "Student Name and ID" }{TEXT 402 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 " Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 257 5 "M APLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" } {TEXT -1 28 ", you will be asked to use " }{TEXT 310 5 "MAPLE" } {TEXT -1 75 " to plot direction fields of first order differential eq uations. The main" }{TEXT 311 7 " MAPLE" }{TEXT -1 32 " function t hat will be used is" }{TEXT 258 2 " " }{TEXT 312 10 "dfieldplot" } {TEXT -1 48 ". Information on, and examples of, the use of " }{TEXT 259 10 "dfieldplot" }{TEXT -1 45 " may be found in the (read-only) wo rksheet " }{HYPERLNK 17 "1.3epR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 2 ". " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }}{PARA 0 "" 0 " " {TEXT -1 30 "Reports that you prepare with " }{TEXT 313 5 "MAPLE" } {TEXT -1 112 " should be prepared with the same care that you would \+ devote to laboratory reports in biology and chemistry. " }{TEXT 315 46 "A report should not be a diary or history of a" }{TEXT -1 1 " " } {TEXT 314 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 316 1 " " }{TEXT 317 51 " session. Delete what is not needed for the report." }{TEXT -1 25 " A ll lines of the form " }{TEXT 281 6 "?topic" }{TEXT -1 82 " (that ar ise from help queries) should be erased. All errors should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Whe n you are printing a " }{TEXT 318 5 "MAPLE" }{TEXT -1 165 " report, t hink about the toner and paper resources that you are using. All comma nds must be terminated - either with the standard terminator, the semi colon, or the " }{TEXT 322 17 "silent terminator" }{TEXT -1 57 ", the \+ colon. When you assign a variable, for example \n \"" }{TEXT 360 7 "x := 5;" }{TEXT -1 30 " \", there is no need to have " }{TEXT 319 5 "M APLE" }{TEXT -1 12 " echo back " }{TEXT 282 6 "x := 5" }{TEXT -1 89 " . When this is printed, it simply wastes paper and ink. Choose the s ilent terminator \"" }{TEXT 283 7 "x := 5:" }{TEXT -1 70 " \" instea d. When you load a package (without the silent terminator)," }{TEXT 284 1 " " }{TEXT -1 2 " " }{TEXT 320 5 "MAPLE" }{TEXT -1 123 " will l ist the commands that become available with the package. This is fine \+ - it will help you become familiar with what " }{TEXT 321 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 text in this worksheet should be deleted. For example, delete \+ the " }{TEXT 288 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 289 8 "K eywords" }{TEXT -1 34 " sections. Delete this section on " }{TEXT 290 7 "Reports" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your worksheet should execute in t he order that it has been written. In particular, remember that the d itto refers to the result of the last executed command - not the resul t of the command that physically precedes it. " }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 8 "Keywords" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {HYPERLNK 17 "color" 2 "color" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "DE plot" 2 "DEplot" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "DEtools" 2 "DEto ols" "" }{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 3 ", " }{HYPERLNK 17 "unapply" 2 "unapply" "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Background Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 420 "The following worksheets, available for download from the syllabus web page, have examples or discussions that will h elp you do this homework. If they are in the same directory as this w orksheet, and if you have retained the filename under which they were \+ posted, then clicking on the hyperlink below will automatically open t hem. Use the Window menu to control the view when multiple files are o pened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{HYPERLNK 17 "1.1epR4.mws" 1 "1.1epR4.mws" " " }{TEXT -1 8 " , " }{HYPERLNK 17 "1.3epR4.mws" 1 "1.3epR4.mws" " " }{TEXT -1 6 " , " }{HYPERLNK 17 "Tutor1R4.mws" 1 "Tutor1R4.mws" " " }}}{SECT 1 {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 323 18 "Backgroud Reading:" }{TEXT -1 21 " The s ection titled \"" }{TEXT 324 30 "A Simple Differential Equation" } {TEXT -1 17 "\" of worksheet " }{HYPERLNK 17 "1.1epR4.mws" 1 "1.1epR 4.mws" "" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 " \+ The help pages for " }{HYPERLNK 17 "subs " 2 "subs" "" }{TEXT -1 8 " and " }{HYPERLNK 17 "testeq" 2 "testeq " "" }{TEXT -1 36 ". (See also the help page for " }{HYPERLNK 17 "odetest" 2 "odetest" "" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Use " }{TEXT 294 5 "MAPLE" } {TEXT -1 8 " to " }{HYPERLNK 17 "verify" 2 "testeq" "" }{TEXT -1 9 " that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 26 " " }{TEXT 295 15 "y(x) = exp(1/x)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "is a solu tion of the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {TEXT 296 8 " " }{XPPEDIT 297 1 "x^4 * diff( y(x), x,x) + 2x^3 \+ *diff(y(x),x) - y(x) = 0" "6#/,(*&%\"xG\"\"%-%%diffG6%-%\"yG6#F&F&F&\" \"\"F.*(\"\"#F.*$F&\"\"$F.-F)6$-F,6#F&F&F.F.-F,6#F&!\"\"\"\"!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 1 " " }} {PARA 0 "" 0 "" {TEXT -1 179 "Do this by replacing the question marks \+ in the next two execution groups with appropriate arguments. Execute e ach group 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 12 "subs(? , ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "testeq( ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 365 7 "MAPLE 6" }{TEXT -1 2 " " } {TEXT 367 5 "Note:" }{TEXT -1 2 " " }{TEXT 369 14 "Beginning with" } {TEXT -1 2 " " }{TEXT 366 11 "MAPLEV R5 " }{TEXT 370 33 "users have \+ been provided with the" }{TEXT -1 2 " " }{TEXT 368 7 "odetest" } {TEXT -1 2 " " }{TEXT 372 14 "command. If " }{TEXT 371 3 "ode" } {TEXT 382 33 " is a differential equation for " }{TEXT 373 6 " y(x) \+ " }{TEXT 383 9 " and if " }{TEXT 374 4 " eqn" }{TEXT 384 31 " is an \+ equation that involves " }{TEXT 375 4 "y(x)" }{TEXT 385 16 " , that is , if " }{TEXT 376 3 "eqn" }{TEXT 386 56 " is an equation that expli citly or implicitly defines " }{TEXT 377 4 "y(x)" }{TEXT 387 19 " , t hen the call " }{TEXT 378 17 "odetest(eqn, ode)" }{TEXT 388 20 " te sts to see if " }{TEXT 379 4 "y(x)" }{TEXT 389 20 " is a solution of " }{TEXT 380 3 "ode" }{TEXT 390 1 "." }}{PARA 0 "" 0 "" {TEXT 396 16 "The return is " }{TEXT 397 1 "0" }{TEXT 398 7 " if " }{TEXT 391 4 "y(x)" }{TEXT 394 20 " is a solution of " }{TEXT 392 3 "ode" } {TEXT 395 1 "." }{TEXT -1 10 " Try it!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(DEtools):\nodetest(?, ?);" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 259 "" 0 "" {TEXT -1 36 "Exercise 2 Free Fall Under Gr avity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 325 18 "Backgroud Reading:" }{TEXT -1 14 " Worksheet " }{HYPERLNK 17 "1.3e pR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 " The help pages for " } {HYPERLNK 17 "dfieldplot" 2 "DEtools[dfieldplot]" "" }{TEXT -1 8 " an d " }{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 -1 124 "If an object is subject only to the force of the Earth's gravity, then the differential equation that relates its \+ velocity " }{TEXT 299 1 "v" }{TEXT 302 1 " " }{TEXT -1 14 " (measured in " }{TEXT 298 3 "m/s" }{TEXT -1 21 ") and its distance " }{TEXT 300 1 "r" }{TEXT -1 43 " from the earth's center in (measured in " } {TEXT 301 2 "m)" }{TEXT -1 4 ", is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{TEXT 326 3 " " }{TEXT 329 3 " " } {TEXT 328 8 " " }{XPPEDIT 327 1 "diff(v(r),r) = - g*R^2/v(r)/r^ 2" "6#/-%%diffG6$-%\"vG6#%\"rGF*,$**%\"gG\"\"\"*$%\"RG\"\"#F.-F(6#F*! \"\"*$F*F1F4F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Here " }{TEXT 304 1 "g " }{TEXT 308 1 " " }{TEXT 305 6 "= 9.81" }{TEXT 303 6 " m/s/s" }{TEXT -1 10 " and " }{TEXT 306 13 " R = 6378000" }{TEXT -1 1 " " } {TEXT 307 1 "m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "a ) Direction Field Plot" }}{PARA 0 "" 0 "" {TEXT -1 71 "Plot the direct ion field of this differential equation in the window " }{TEXT 291 39 "[ 6378000 , 20000000 ] x [ 50 , 12000 ]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 122 "Do this by replacing the question marks with a ppropriate arguments. For the last execution group, look up the avail able " }{HYPERLNK 17 "color choices" 2 "color" "" }{TEXT -1 105 " an d pick one you like. (Light colors do not alwayys print out well.) E xecute each line in succession." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart: with(plots): wi th(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ode := ? ; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := 9.81: R := 63780 00:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dfieldplot(?, ?, r = 6378000 .. 20000000, v = 50 .. 12000, color = ? );" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Before continuing, choose a name and assign the plot st ructure that you have just created to that name." }}{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 1 {PARA 4 "" 0 "" {TEXT -1 53 " b) Plotting the Solution of One Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 42 "If the initial velocity of an object is " }{TEXT 261 18 "v(6378000) = 12000" }{TEXT 309 1 " " }{TEXT 260 4 "m/s," } {TEXT -1 8 " use " }{TEXT 276 6 "dsolve" }{TEXT -1 35 " to find an explicit formula for " }{TEXT 275 4 "v(r)" }{TEXT -1 10 ". Plot \+ " }{TEXT 264 4 "v(r)" }{TEXT -1 8 " for " }{TEXT 263 1 "r" }{TEXT -1 18 " in the interval " }{TEXT 262 22 "[ 6378000 , 20000000 ]" } {TEXT -1 24 ". Use the plot option " }{TEXT 270 13 "thickness = 2" } {TEXT -1 9 ". (See " }{HYPERLNK 17 "plot,options" 2 "plot,options" " " }{TEXT -1 3 ". )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "soln1 := dso lve( \{?, ?\}, v(r) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v 1 := unapply(rhs(soln1), r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot(?, r = 6378000 .. 20000000, thickness = 2, color = ?);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Before c ontinuing, name the plot structure that you just created." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := % :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "c) Plotting the Solution of a S econd Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 42 "If the ini tial velocity of an object is " }{TEXT 266 17 "v(6378000) = 8000" } {TEXT 330 1 " " }{TEXT 265 4 "m/s," }{TEXT -1 7 " use " }{TEXT 278 6 "dsolve" }{TEXT -1 35 " to find an explicit formula for " }{TEXT 277 4 "v(r)" }{TEXT -1 10 ". Plot " }{TEXT 269 4 "v(r)" }{TEXT -1 8 " for " }{TEXT 268 1 "r" }{TEXT -1 18 " in the interval " } {TEXT 267 22 "[ 6378000 , 13000000 ]" }{TEXT -1 25 ". Use the plot o ption " }{TEXT 271 13 "thickness = 2" }{TEXT -1 8 ". See " } {HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "so ln2 := dsolve( \{?, ?\}, ? );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v2 := unapply(?, ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(?, r = ?, thickness = 2, color = ?); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Before continuing, name the plot structure that you just create d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := %:" }}}{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 27 "d) Superimposing Every thing" }}{PARA 0 "" 0 "" {TEXT -1 51 "Superimpose the plots that you \+ obtained in parts " }{TEXT 272 2 "a)" }{TEXT -1 2 ", " }{TEXT 273 2 " b)" }{TEXT -1 6 ", and " }{TEXT 274 2 "c)" }{TEXT -1 77 ". (This will look nicest 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 19 "display(\{?, ?, ?\});" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 68 "Exercise 3 Direction Field Project (Adapted From \+ Edwards and Penney)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 363 18 "Backgroud Reading:" }{TEXT -1 14 " Worksheet " } {HYPERLNK 17 "1.3epR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 59 " Th e help pages for " }{HYPERLNK 17 "dfieldplot" 2 "DEtools[dfieldplot] " "" }{TEXT -1 5 " , " }{HYPERLNK 17 "display" 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 346 5 "MAPLE" }{TEXT -1 22 " ker nel and load the " }{TEXT 347 7 "DEtools" }{TEXT -1 68 " package. (Pla ce the cursor anywhere in the next line and press the " }{TEXT 348 5 " Enter" }{TEXT -1 6 " key.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: with(DEtools):" }}} {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 37 "ode : = diff(y(x), x) = sin(x - y(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 110 "Don't forget to execute the preceding ex ecution group (by placing the cursor anywhere within and pressing the \+ " }{TEXT 349 5 "Enter" }{TEXT -1 6 " key.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "a) Using the Direction 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 351 10 "dfieldplot" }{TEXT -1 76 " to plot the direction field of the given equation in the viewing win dow " }{TEXT 352 21 "[-10, 10] 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 shou ld 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 substitute " }{TEXT 353 14 "y(x) = \+ m*x + b" }{TEXT -1 57 " into the given equation. Then simplify the e quation (" }{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 12 "simplify(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "You will see that the lef t hand side is constant. The right hand side can therefore not depend \+ on " }{TEXT 354 1 "x" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 36 "What can you conclude the value of " }{TEXT 355 1 "m" }{TEXT -1 5 " is?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m := 1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "solve" 2 "solve" "" }{TEXT -1 7 " or " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 80 " (w ith the three parameters: equation, variable, range) to find the valu e of " }{TEXT 356 1 "b" }{TEXT -1 25 " that is least negative." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fsolve( -arcsin(1) = b, b, - 2 .. 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( 1 = - si n(b), b);" }}}{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 23 "b) The General Solution" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " } {TEXT 357 5 "MAPLE" }{TEXT -1 59 " to find a general solution to the differential 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 350 5 "MAPLE" }{TEXT -1 31 " returns is implicit but 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 27 "eqn := y(x) = solve(? , ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 57 "One particular solution of the differenti al 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 cons tant " }{MPLTEXT 1 0 3 "_C1" }{TEXT -1 75 " that corresponds to th is solution (or any other straight line solution)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "However, plot the solutio ns that correspond 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 361 10 " " }{XPPEDIT 362 1 "limit(x-2*arctan((-2+x+_C1)/(x+_C1)),_C1=infinity)" "6#-%&limit G6$,&%\"xG\"\"\"*&\"\"#F(-%'arctanG6#*&,(F*!\"\"F'F(%$_C1GF(F(,&F'F(F1 F(F0F(F0/F1%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 345 5 "MAPLE" }{TEXT -1 12 " Functio n " }{TEXT 344 6 "DEplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 364 18 "Backgroud Reading:" }{TEXT -1 29 " None. (The h elp page for " }{HYPERLNK 17 "DEplot" 2 "DEtools[DEplot]" "" }{TEXT -1 149 " is fairly rough going. Of course you may look at it if you \+ wish but you will probably be better off just reading and emulating th e example below.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 47 "Be sure to execute the group \+ before continuing!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 4 " " }{TEXT 331 44 " y'(x) = 1/(1+x^2) + sin(y(x)), y( -5) = 6:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "restart: with(DEtools): with(plots):\node := diff(y (x), x) = 1/(1+x^2) + sin(y(x)): \nDEplot(ode , [y(x)] , x = -5 .. 10, [[ y(-5)=6 ]] );" }}}{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 abo ve 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 334 1 "x" }{TEXT -1 15 " increases t o " }{TEXT 332 3 " 10" }{TEXT -1 8 ". Use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 42 " to study the behavior for values of \+ " }{TEXT 333 1 "x" }{TEXT -1 9 " between " }{MPLTEXT 1 0 3 "150" } {TEXT -1 6 " and " }{MPLTEXT 1 0 3 "200" }{TEXT -1 44 ". Describe wh at appears to be happening to" }{TEXT 336 5 " y(x)" }{TEXT -1 6 " as " }{TEXT 335 1 "x" }{TEXT -1 20 " tends to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "DEplot(? , ? , ?, ?);" }}}{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 35 "b) Two Other Initial Value Problems" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 49 " to study \+ solutions to the Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 286 44 "y'(x) = 1/(1+x^2) + sin(y(x)), y(- 5) = - 6" }}{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 287 45 "y'(x) = 1/(1+x^2) + sin(y(x)), y(- 5) = - 15" }{TEXT 337 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{TEXT 343 1 "x" }{TEXT -1 11 " between " }{TEXT 339 2 "-5" }{TEXT -1 7 " and \+ " }{TEXT 338 2 "20" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "DEplot(? , ? , ?, ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "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 ate the possible" }}{PARA 0 "" 0 "" {TEXT -1 9 " values of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " " }{TEXT 340 4 " " }{XPPEDIT 341 1 "l imit(y(x),x=infinity)" "6#-%&limitG6$-%\"yG6#%\"xG/F)%)infinityG" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "if " } {TEXT 342 4 "y(x)" }{TEXT -1 191 " satisfies the given equation? (C lick on the curves to guess this limit. For the first you should be a ble to see what exact number the limit is. Use that value as a hint f or 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 (w ithout having an explicit formula) " }}{PARA 0 "" 0 "" {TEXT -1 74 " U se the differential equation to explain the reason for the behavior of " }{TEXT 359 4 "y(x)" }{TEXT -1 6 " as " }{TEXT 358 1 "x" }{TEXT -1 73 " tends to infinity. (You should aim for plausibility rather th an 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 Author Information" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 36 "01F00R6.mws A Maple 6 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 44 "Author: Brian E. Blank (06 September 2000)" }}{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, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permi ssion of the author." }}{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 " Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " \+ Washington University in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ St. Louis, MO 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 "Copyr ight: \251 2000 Brian E. Blank, All Rights Reserved." }}}}}{MARK "1 \+ 0 0" 260 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }