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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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 365 20 "Student Name and ID :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 366 20 "Stu dent Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" "" }{TEXT -1 28 ", yo u will be asked to use " }{TEXT 310 5 "MAPLE" }{TEXT -1 75 " to plot \+ direction fields of first order differential equations. The main" } {TEXT 311 7 " MAPLE" }{TEXT -1 32 " function that will be used is" }{TEXT 258 2 " " }{TEXT 312 10 "dfieldplot" }{TEXT -1 48 ". Informat ion on, and examples of, the use of " }{TEXT 259 10 "dfieldplot" } {TEXT -1 45 " may be found in the (read-only) worksheet " } {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 repo rt 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 " All lines o f the form " }{TEXT 281 6 "?topic" }{TEXT -1 82 " (that arise from h elp queries) should be erased. All errors should be erased. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "When you \+ are printing a " }{TEXT 318 5 "MAPLE" }{TEXT -1 165 " report, think a bout the toner and paper resources that you are using. All commands mu st be terminated - either with the standard terminator, the semicolon, 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 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 282 6 "x := 5" }{TEXT -1 89 ". Whe n this is printed, it simply wastes paper and ink. Choose the silent \+ terminator \"" }{TEXT 283 7 "x := 5:" }{TEXT -1 70 " \" instead. Wh en you load a package (without the silent terminator)," }{TEXT 284 1 " " }{TEXT -1 2 " " }{TEXT 320 5 "MAPLE" }{TEXT -1 123 " will list the commands that become available with the package. This is fine - it wi ll help you become familiar with what " }{TEXT 321 5 "MAPLE" }{TEXT -1 125 " makes available. However, these commands should not be part o f a lab report. Reload the package with the silent terminator. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Much of t he text in this worksheet should be deleted. For example, delete the \+ " }{TEXT 288 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 289 8 "Keywo rds" }{TEXT -1 34 " sections. Delete this section on " }{TEXT 290 7 "R eports" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your worksheet should execute in the \+ order that it has been written. In particular, remember that the ditt o refers to the result of the last executed command - not the result o f 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 "DEplot" 2 "D Eplot" "" }{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 -1 420 "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 "1.3 epR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 6 " , " }{HYPERLNK 17 "Tutor 1R4.mws" 1 "Tutor1R4.mws" "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "E xercise 1 Checking that a Differential Equation is Satisfied" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 18 "Backgrou d Reading:" }{TEXT -1 21 " The section titled \"" }{TEXT 324 30 "A Sim ple Differential Equation" }{TEXT -1 17 "\" of worksheet " } {HYPERLNK 17 "1.1epR4.mws" 1 "1.1epR4.mws" "" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 59 " Th e help pages for " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 8 " an d " }{HYPERLNK 17 "testeq" 2 "testeq" "" }{TEXT -1 1 "." }}{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) = e xp(1/x)" }}{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 5 " " }{TEXT 296 8 " " }{XPPEDIT 297 1 "x^4 * diff( y(x) , x,x) + 2x^3 *diff(y(x),x) - y(x) = 0" "/,(*&%\"xG\"\"%-%%diffG6%-%\" yG6#F%F%F%\"\"\"F-*(\"\"#F-*$F%\"\"$F--F(6$-F+6#F%F%F-F--F+6#F%!\"\"\" \"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 179 "Do this by replacing the question ma rks in the next two execution groups with appropriate arguments. Execu te each group by placing the cursor within it and pressing the Enter k ey." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(? , ?);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "testeq( ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 36 "Exercise 2 F ree Fall Under Gravity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 325 18 "Backgroud Reading:" }{TEXT -1 14 " Worksheet " } {HYPERLNK 17 "1.3epR4.mws" 1 "1.3epR4.mws" "" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 59 " Th e help pages for " }{HYPERLNK 17 "dfieldplot" 2 "DEtools[dfieldplot] " "" }{TEXT -1 8 " and " }{HYPERLNK 17 "display" 2 "plots[display] " "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 124 "If an object is subj ect only to the force of the Earth's gravity, then the differential eq uation that relates its velocity " }{TEXT 299 1 "v" }{TEXT 302 1 " " }{TEXT -1 14 " (measured in " }{TEXT 298 3 "m/s" }{TEXT -1 21 ") and \+ its distance " }{TEXT 300 1 "r" }{TEXT -1 43 " from the earth's cent er in (measured in " }{TEXT 301 2 "m)" }{TEXT -1 4 ", is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 34 " " }{TEXT 326 3 " " }{TEXT 329 3 " " }{TEXT 328 8 " " }{XPPEDIT 327 1 "diff(v(r), r) = - g*R^2/v(r)/r^2" "/-%%diffG6$-%\"vG6#%\"rGF),$**%\"gG\"\"\"*$%\" RG\"\"#F--F'6#F)!\"\"*$F)F0F3F3" }}{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 direction 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 questio n marks with appropriate arguments. For the last execution group, loo k up the available " }{HYPERLNK 17 "color choices" 2 "color" "" } {TEXT -1 105 " and pick one you like. (Light colors do not alwayys p rint out well.) Execute each line in succession." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart: w ith(plots): with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ode := ? ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := 9. 81: R := 6378000:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dfie ldplot(?, ?, 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 as sign the plot structure 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 Proble m" }}{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 "th ickness = 2" }{TEXT -1 9 ". (See " }{HYPERLNK 17 "plot,options" 2 "p lot,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 := dsolve( \{?, ?\}, v(r) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v1 := 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 continuing, 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) Plott ing the Solution of a Second Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 42 "If the initial 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 option " }{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 "soln2 := 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 created." }}{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) Sup erimposing Everything" }}{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 t he 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 0 {PARA 260 "" 0 "" {TEXT -1 68 "Exercise 3 Direction Field Proj ect (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 " \+ The 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 " kernel and load the " }{TEXT 347 7 "DEtools" }{TEXT -1 68 " package . (Place 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 "Conside r the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "od e := 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 \+ execution group (by placing the cursor anywhere within and pressing th e " }{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)" "-%&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 0 {PARA 3 "" 0 "" {TEXT -1 17 "Exercise 4 The " }{TEXT 345 5 " MAPLE" }{TEXT -1 12 " Function " }{TEXT 344 6 "DEplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 364 18 "Backgroud Reading: " }{TEXT -1 29 " None. (The help page for " }{HYPERLNK 17 "DEplot" 2 "DEtools[DEplot]" "" }{TEXT -1 149 " is fairly rough going. Of cou rse you may look at it if you wish but you will probably be better off just reading and emulating the 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 the 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 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 54 "PLease re ad and then execute the next execution group." }}{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 := dif f(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 coordinat es above the worksheet, to the left of the plot toolbar." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {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 to " }{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 ". De scribe what 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 0 {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) + s in(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) + si n(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 tha t 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 "limit(y(x),x=infinity)" "-%&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? (Click on the curves to guess this limit. For the f irst you should be able to see what exact number the limit is. Use th at 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 0 {PARA 4 "" 0 "" {TEXT -1 100 "c) Using a Differential Equation to Unde rstand Solution Curves (without having an explicit formula) " }}{PARA 0 "" 0 "" {TEXT -1 74 " Use the differential equation to explain the r eason 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 fo r 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 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 45 "01F00R4.mws A MapleV Release 4 work sheet." }}{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 "includin g print, disk, and electronic transfer, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission 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-ma il: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }} {PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 Brian E. Blank, \+ All Rights Reserved." }}}}}{MARK "0 0 0" 16 }{VIEWOPTS 1 1 0 3 4 1802 }