{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 24 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 255 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 255 0 255 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 255 0 255 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 14 255 0 255 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 128 0 128 1 1 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "Times" 1 18 0 0 255 1 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 287 "Times" 1 18 0 0 255 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 " Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 293 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "Cou rier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 299 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "Cour ier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 308 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 310 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 312 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 313 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "Cour ier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "Courier" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 319 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 321 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 323 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 324 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 325 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 326 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 327 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 19 328 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 332 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 335 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 336 "Courier" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 338 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 0 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 " " 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 18 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 270 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 18 "Laplace transforms" }} {PARA 258 "" 0 "" {TEXT 256 4 "HW 4" }}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 22 "Student Name and ID : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 21 "S tudent 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 ", \+ you will be asked to use " }{TEXT 266 5 "MAPLE" }{TEXT -1 71 " to so lve nonhomogeneous constant coefficient differential equations. " }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " }{TEXT 267 5 "MAPLE" }{TEXT -1 112 " should be prepared with the same care that you would devote to laboratory reports in biology and chemistry. " }{TEXT 269 46 "A repo rt should not be a diary or history of a" }{TEXT -1 1 " " }{TEXT 268 6 " MAPLE" }{TEXT -1 2 " " }{TEXT 270 1 " " }{TEXT 271 51 "session. \+ Delete what is not needed for the report." }{TEXT -1 25 " All lines o f the form " }{TEXT 259 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 272 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 276 17 "silent terminator" }{TEXT -1 57 ", the colon. When you assign a variable, for example \n \"" }{TEXT 278 7 "x := 5; " }{TEXT -1 30 " \", there is no need to have " }{TEXT 273 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 260 6 "x := 5" }{TEXT -1 89 ". Whe n this is printed, it simply wastes paper and ink. Choose the silent \+ terminator \"" }{TEXT 261 7 "x := 5:" }{TEXT -1 70 " \" instead. Wh en you load a package (without the silent terminator)," }{TEXT 262 1 " " }{TEXT -1 2 " " }{TEXT 274 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 275 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 263 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 264 8 "Keywo rds" }{TEXT -1 34 " sections. Delete this section on " }{TEXT 265 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 3 " " } {HYPERLNK 17 "DEplot" 2 "DEtools[DEplot]" "" }{TEXT -1 4 " , " } {HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 5 " , " }{HYPERLNK 17 "Dirac" 2 "Dirac" "" }{TEXT -1 6 ", " }{HYPERLNK 17 "floor" 2 " floor" "" }{TEXT -1 7 " , " }{HYPERLNK 17 "Heaviside" 2 "Heaviside " "" }{TEXT -1 6 " , " }{HYPERLNK 17 "inttrans" 2 "inttrans" "" } {TEXT -1 6 " , " }{HYPERLNK 17 "invlaplace" 2 "invlaplace" "" } {TEXT -1 4 " , " }{HYPERLNK 17 "isolate" 2 "isolate" "" }{TEXT -1 3 " , " }{HYPERLNK 17 "laplace" 2 "laplace" "" }{TEXT -1 5 " , " } {HYPERLNK 17 "map" 2 "map" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "rhs" 2 "rhs" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "sum" 2 "sum" "" }{TEXT -1 5 " , " }{HYPERLNK 17 "unapply" 2 "unapply" "" }{TEXT -1 2 " ," }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Background Worksheets" }}{PARA 0 "" 0 "" {TEXT -1 420 "Th e following worksheets, available for download from the syllabus web \+ page, have examples or discussions that will help you do this homework . If they are in the same directory as this worksheet, and if you hav e retained the filename 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 2 " " } {HYPERLNK 17 "4.1-4.6epR4.mws" 1 "4.1-4.6epR4.mws" "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "Exercise 1 T he Laplace Transform of a Piecewise Defined Function" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 18 "Backgroud Reading:" } {TEXT -1 5 " " }{HYPERLNK 17 "4.1-4.6epR4.mws" 1 "4.1-4.6epR4.mws " "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "a) Consider the IVP " }{XPPEDIT 19 1 "diff(x(t),t,t) + diff( x(t),t) + 4*x(t) = t,x(0)=2,D(x)(0)=0" "6%/,(-%%diffG6%-%\"xG6#%\"tGF +F+\"\"\"-F&6$-F)6#F+F+F,*&\"\"%F,-F)6#F+F,F,F+/-F)6#\"\"!\"\"#/--%\"D G6#F)6#F8F8" }}{PARA 0 "" 0 "" {TEXT -1 45 "Name the differential equa tion in this I.V.P." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "restart; with(inttrans); # Execute this li ne!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "? := diff(x(t),t$2) \+ + diff(x(t),t) + 4*x(t) = t;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "Apply the Laplace transform to it and nam e the resulting equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "? := map( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Isolate the Laplace \+ transform of " }{MPLTEXT 1 0 4 "x(t)" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "readlib (isolate); # Remove this line in R6 - there is no need to readlib in R 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "? := isolate( ? , ? ); " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 30 "Su bstitute the initial values:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "su bs( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Apply the inverse Laplace transform:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "map( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 9 "Define " }{MPLTEXT 1 0 4 "x(t)" } {TEXT -1 3 ": \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "x := ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Plot " }{MPLTEXT 1 0 4 "x(t)" }{TEXT -1 8 " for " }{MPLTEXT 1 0 11 " t= 0 .. 25" }{TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(x(t), t = 0.. 12, linestyle = \+ 2); # Execute this line" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 15 "Name your plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "plot1 := \":" }}}{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) Calculate the \+ Laplace Transform " }{XPPEDIT 19 1 "F(s" "-%\"FG6#%\"sG" }{TEXT -1 7 " of " }{XPPEDIT 19 1 "f(t) = PIECEWISE([t, t < 9],[9, 9-t < 0 and t -12 < 0],[45-3*t, 12-t < 0 and t-15 < 0],[0, 15 < t])" "/-%\"fG6#%\"tG -%*PIECEWISEG6&7$F&2F&\"\"*7$F,32,&F,\"\"\"F&!\"\"\"\"!2,&F&F1\"#7F2F3 7$,&\"#XF1*&\"\"$F1F&F1F232,&F6F1F&F2F32,&F&F1\"#:F2F37$F32FAF&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7co7$\"\"!F(7 $$\"1LLLL3VfV!#;F*7$$\"1nmm\"H[D:)F,F.7$$\"1LLLe0$=C\"!#:F17$$\"1MLL3R Br;F3F57$$\"1nm;zjf)4#F3F87$$\"1LL$e4;[\\#F3F;7$$\"1++]i'y]!HF3F>7$$\" 1ML$ezs$HLF3FA7$$\"1++]7iI_PF3FD7$$\"1nmm;_M(=%F3FG7$$\"1MLL3y_qXF3FJ7 $$\"1,++]1!>+&F3FM7$$\"1+++]Z/NaF3FP7$$\"1+++]$fC&eF3FS7$$\"1ML$ez6:B' F3FV7$$\"1nmm;=C#o'F3FY7$$\"1nmmm#pS1(F3Ffn7$$\"1++]i`A3vF3Fin7$$\"1mm mm(y8!zF3F\\o7$$\"1,+]i.tK$)F3F_o7$$\"1,+](3zMu)F3Fbo7$$\"1o;a)QA1&))F 3Feo7$$\"1NLe*olx&*)F3Fho7$$\"1^P%[^^X)*)F3F[p7$$\"1oT5StL6!*F3$\"\"*F (7$$\"1&ek`;B\"Q!*F3F`p7$$\"1-]i!**3\\1*F3F`p7$$\"1Ne9T1[=\"*F3F`p7$$ \"1omm\"H_?<*F3F`p7$$\"1omT&GM)o$*F3F`p7$$\"1nm;zihl&*F3F`p7$$\"1LLL3# G,***F3F`p7$$\"1LLezw5V5!#9F`p7$$\"1++v$Q#\\\"3\"FjqF`p7$$\"1LL$e\"*[H 7\"FjqF`p7$$\"1nm\"HCjV9\"FjqF`p7$$\"1+++qvxl6FjqF`p7$$\"1+]iSCDw6FjqF `p7$$\"1++D6ts'=\"FjqF`p7$$\"1+DcYZ'>>\"FjqF`p7$$\"1+](==-s>\"FjqF`p7$ $\"1+v=<'RC?\"Fjq$\"1%*\\P%[6o#*)F37$$\"1++]_qn27Fjq$\"1)****\\U)op()F 37$$\"1+]P/q%zA\"Fjq$\"1***\\(o)*eh\")F37$$\"1++Dcp@[7Fjq$\"1++]78\\`v F37$$\"1+](=GB2F\"Fjq$\"1)**\\Pa,$yoF37$$\"1++]2'HKH\"Fjq$\"1'****\\x6 J?'F37$$\"1LL3UDX88Fjq$\"1/+]PPU'f&F37$$\"1nmmwanL8Fjq$\"1/+++dt*)\\F3 7$$\"1LL$exn_N\"Fjq$\"1*****\\sm>M%F37$$\"1+++v+'oP\"Fjq$\"1,++]x>%p$F 37$$\"1n;H2fU'R\"Fjq$\"1-+D\"yAs5$F37$$\"1LLeR<*fT\"Fjq$\"1.+]7yC?DF37 $$\"1n;HiBQP9Fjq$\"1)**\\78H&y=F37$$\"1+++&)Hxe9Fjq$\"1$*****\\/\"oB\" F37$$\"1L$eR*)**)y9Fjq$\"1R+]7=.IjF,7$$\"1nm\"H!o-*\\\"Fjq$\"1`++]7f>H !#<7$$\"1LL3A_1?:FjqF(7$$\"1++DTO5T:FjqF(7$$\"1nmmT9C#e\"FjqF(7$$\"1++ D1*3`i\"FjqF(7$$\"1MLL$*zym;FjqF(7$$\"1ML$3N1#4FjqF(7$$\"1++v.Uac>FjqF(7$$\"#?F(F(-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Fiz;F ($F`[lF(" 2 812 333 333 2 0 1 2 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10072 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 314 42 0 0 0 0 0 0 }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 55 "Using the definition of the Laplace transform, def ine " }{MPLTEXT 1 0 1 "F" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "F := s -> ? ;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "c) Solve the Initial Value Problem " }{XPPEDIT 19 1 "diff(x(t),t,t) + diff(x(t),t) + 4*x(t) = f(t),x(0)=2, D(x)(0)=0" "6%/,(-%%diffG6%-%\"xG6#%\"tGF+F+\"\"\"-F&6$-F)6#F+F+F,*&\" \"%F,-F)6#F+F,F,-%\"fG6#F+/-F)6#\"\"!\"\"#/--%\"DG6#F)6#F;F;" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Restore the initial unassigned literal value to " }{MPLTEXT 1 0 1 "x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x := 'x': # Execute this line!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Name the new differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "? := diff(x(t),t,t )+diff(x(t),t)+4*x(t) = f(t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "Apply the Laplace transform to it as foll ows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "map( ? , lhs(?)) = F(s);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Substi tute the initial values:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "? := subs( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Isolate the Laplace \+ transform:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "? := isolate( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Apply the inverse Laplace transform:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "map( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Define this " }{MPLTEXT 1 0 1 "x" }{TEXT -1 1 ":" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x := t -> " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Plot the solution you hav e found for " }{TEXT 315 10 "0 < t < 25" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(x(t), t = 0 .. 25, color = navy); # E xecute this line" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Name the plot:" }}{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 -1 111 "Superimpose this plot on that of the graph of the solution of the initial value problem discussed in Part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(?,?);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 68 "Any (com mon sense) principle to be discerned from the superposition?" }}{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 259 "" 0 "" {TEXT -1 60 "Exercise 2 Saw-Too th, Triangular-Wave, Square-Wave Input" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 18 "Backgroud Reading:" }{TEXT -1 4 " " }{HYPERLNK 17 "4.1-4.6epR4.mws " 1 "4.1-4.6epR4.mws " "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "a ) Greatest Integer Function or Floor Function or Staircase F unction" }}{PARA 4 "" 0 "" {TEXT -1 57 "\n(Must be a great function if it has three names, right?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The greatest integer function of " }{TEXT 286 1 "x" }{TEXT -1 65 " is defined to be the greatest integer that is no t larger than " }{TEXT 287 1 "x" }{TEXT -1 18 ". It is denoted " } {TEXT 285 5 "floor" }{TEXT -1 6 " in " }{TEXT 284 5 "MAPLE" }{TEXT -1 7 " . Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "restart; wit h(inttrans): with(plots): #Execute this line and the next!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "'floor( Pi )' = floor( Pi );\n'flo or( - Pi )' = floor( - Pi );\n'floor(- 7 )' = floor(- 7 );\n'floor( 13 )' = floor( 13 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "\nThe plot \+ of " }{TEXT 288 9 " floor(t)" }{TEXT 316 1 " " }{TEXT -1 62 " expla ins why it is sometimes called the staircase function:\n" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The current releas e of " }{TEXT 292 5 "MAPLE" }{TEXT -1 44 " does not know the Lapla ce Transform of " }{TEXT 293 8 "floor(t)" }{TEXT -1 47 ". But it d oes know the Laplace Transform of " }{TEXT 294 12 "Heaviside(t)" } {TEXT -1 58 " and we can use that to derive the Laplace Transform of \+ " }{TEXT 295 8 "floor(t)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "laplace(Heaviside(t), t, s); # Execute this line - bu t you should already know the result!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "Express " }{TEXT 289 8 "floor(t)" } {TEXT -1 70 " in terms of an infinite sum of translates of the Heavis ide function " }{TEXT 290 1 " " }{TEXT 296 12 "Heaviside(t)" }{TEXT -1 69 " . Use this representation to calculate the Laplace transform \+ of " }{TEXT 291 8 "floor(t)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Using your own name, one \+ that has no a priori meaning to " }{TEXT 318 5 "MAPLE" }{TEXT -1 63 ", define the floor function in terms of the Heaviside function:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "? := t -> sum( ? , ? );" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Apply the Laplace transform to compute the Lap lace transform of the floor function. Name the expression." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "? \+ := laplace( ? , ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 103 "This expression can be used to define the functio n that is the Laplace transform of the floor function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "? = unapply( ? , ? );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "b ) Three Engineering Functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "saw_tooth := t -> 2*t-2*floor(t)-1; # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "triangular_wave := t -> abs(2*t+1-4*floor (1/2*t+3/4))-1; # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "square_wave := t -> signum(abs(2*t+1-4*floor(1/2*t+3/4))-1); # E xecute!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Here are the plots of these three \+ functions for " }{MPLTEXT 1 0 10 "t = 0 .. 6" }{TEXT -1 26 ", as ty pically rendered:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot( saw_tooth(t), t = 0 .. 6); # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot( triangular_wave(t) \+ , t = 0 .. 6); # Execute !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot( square_wave(t) , t = 0 .. 6); # Execute!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 72 "Calculate the Laplac e Transforms of these three \"engineering functions\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 5 "Hint:" }{TEXT -1 68 " Use the result of part (a) to calculate the Laplace Transform of \+ " }{TEXT 298 9 "saw_tooth" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "For the two others, use the theorem on Laplace transforms of perio dic functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 319 8 "Sawtooth" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "laplace( saw_tooth(t), t , s ); # Execute !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs( ? , ? );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "`laplace( saw_tooth(t), t , \+ s )` = ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "For the record, let us write it out!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 15 "Triangular Wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "To do the integrati on, you will need to divide the interval [0,p] into three subintervals . Here p denotes the period of the triangular wave.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "`laplace(triangular_wave(t),t,s)` = (int( ?,?) + int(?,?) + int(?,?))/(?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 300 11 "Square Wave" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "In a similar way:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "`laplace(square_wave(t),t,s)` = (in t(?,?) + int(?,?))/(?);" }}}{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 0 "" 0 " " {TEXT 302 7 " c) Use" }{TEXT -1 2 " " }{TEXT 301 6 "DEplot" }{TEXT -1 2 " " }{TEXT 303 51 "to plot the solution to the initial value pro blem " }{XPPEDIT 323 1 "diff(diff(x(t),t),t)+2*diff(x(t),t)+5*x(t) = \+ 3*w(t)*` `, ` `*x(0) = `0.0`*` `, ` `*D(x)(0) = 0" "6%/,(-%% diffG6$-F&6$-%\"xG6#%\"tGF-F-\"\"\"*&\"\"#F.-F&6$-F+6#F-F-F.F.*&\"\"&F .-F+6#F-F.F.*(\"\"$F.-%\"wG6#F-F.%$~~~GF./*&%%~~~~GF.-F+6#\"\"!F.*&%$0 .0GF.FAF./*&F>F.--%\"DG6#F+6#FDF.FD" }{TEXT 322 1 ":" }}{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 29 "in each of the three cases: " }{TEXT 304 1 " " }{TEXT 335 19 " w(t) = saw_tooth(t)" }{TEXT 336 2 ", " }{TEXT 305 25 "w(t) = triangula r_wave(t)" }{TEXT -1 9 ", and " }{TEXT 306 1 " " }{TEXT 338 21 "w(t ) = square_wave(t)" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(DEtools): # Execute" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Here is one to get you started:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(DEtools): # Execute" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 85 "saw_tooth_forcing_eqn := diff(x(t), t$2) + 2*diff(x (t), t) + 5*x(t) = 3*saw_tooth(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "DEplot(\{saw_tooth_forcing_eqn\}, [x(t)], t=0.. 10, [ [x(0)=0.0,D(x)(0)=0]], stepsize=.1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "There remain the triangular and squar e wave plots:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT -1 41 "Exercise 3 Laplace Transforming Systems" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 18 "Backgroud Reading:" } {TEXT -1 4 " " }{HYPERLNK 17 "4.1-4.6epR4.mws" 1 "4.1-4.6epR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 32 "Consider the first order system " } {TEXT 307 1 " " }{XPPEDIT 308 1 "diff(x(t),t) = -3/10*x(t)+y(t)/20,dif f(y(t),t) = 3/10*x(t)-3/20*y(t)" "6$/-%%diffG6$-%\"xG6#%\"tGF*,&*(\"\" $\"\"\"\"#5!\"\"-F(6#F*F.F0*&-%\"yG6#F*F.\"#?F0F./-F%6$-F56#F*F*,&*(F- F.F/F0-F(6#F*F.F.*(F-F.F7F0-F56#F*F.F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Suppose that" }{TEXT 309 2 " " }{XPPEDIT 310 1 "x(0)=15 " "/-%\"xG6#\"\"!\"#:" }{TEXT -1 7 " and " }{TEXT 311 1 " " } {XPPEDIT 312 1 "y(0)=20" "/-%\"yG6#\"\"!\"#?" }{TEXT -1 76 ". (This \+ is an initial value problem for the brine example in Section 5.1.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "Apply th e Laplace transform to each equation. Solve simultaneously For each La place transform. Solve the system by finding the inverse Laplace trans forms. Plot your solutions for " }{TEXT 314 1 " " }{TEXT 313 10 "0 < \+ t < 30" }{TEXT -1 86 " (both in one window). Describe what the behavio r is and why the solution makes sense." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart; with(inttrans) ; # Execute me!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Name the differential equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "? := diff(x( t),t) = -3*x(t)/10+ y(t)/20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "? := diff(y(t),t) = 3*x(t)/10-3*y(t)/20;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Apply the Laplace transform to them:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "? := map( ? , ? ); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Si mplify the notation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " ? := subs (\{laplace(x(t),t,s) = X(s), laplace(y(t),t,s) = Y(s)\}, \{ ? , ? \}); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 57 "Substitute the initial values into this s et of equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "? := subs( \{ ? , ? \} , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Solve the equations for " }{MPLTEXT 1 0 4 "X(s)" }{TEXT -1 8 " and " }{MPLTEXT 1 0 4 "Y(s)" }{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 15 "solve( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "A pply the inverse Laplace transform:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "map( ? , ? );" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Convert from a set to a list (for the sake of definiteness)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "? := convert(\",li st); # Execute this line" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 36 "Let's get just the right hand sides:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "? := map(z -> rhs(z), ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Plot " }{MPLTEXT 1 0 4 "x(t)" }{TEXT -1 7 " and " }{MPLTEXT 1 0 4 "y(t)" }{TEXT -1 8 " \+ for " }{MPLTEXT 1 0 10 "t = 0.. 30" }{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 41 "plot( ? , t \+ = 0.. 30, color = [ ? , ?] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 320 70 "What (common sense) conclusions about th e system do the plots confirm?" }}{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 12 "Exerci se 4 " }{TEXT 282 1 " " }{TEXT -1 9 "Impulses " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 334 18 "Backgroud Reading:" } {TEXT -1 4 " " }{HYPERLNK 17 "4.1-4.6epR4.mws " 1 "4.1-4.6epR4.mw s " "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Consider the differential equations " }{TEXT 324 2 " " }{XPPEDIT 325 1 "diff(diff(x(t),t),t)+2*diff(x(t),t)+5*x(t) = cos(3*t)" "/,(-%%d iffG6$-F%6$-%\"xG6#%\"tGF,F,\"\"\"*&\"\"#F--F%6$-F*6#F,F,F-F-*&\"\"&F- -F*6#F,F-F--%$cosG6#*&\"\"$F-F,F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "and " } {TEXT 326 3 " " }{XPPEDIT 327 1 "diff(diff(x(t),t),t)+2*diff(x(t),t) +5*x(t) = cos(3*t)+3*Dirac(t-5.2)" "/,(-%%diffG6$-F%6$-%\"xG6#%\"tGF,F ,\"\"\"*&\"\"#F--F%6$-F*6#F,F,F-F-*&\"\"&F--F*6#F,F-F-,&-%$cosG6#*&\" \"$F-F,F-F-*&F=F--%&DiracG6#,&F,F-$\"#_!\"\"FEF-F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "with initial values" } {TEXT 330 2 " " }{XPPEDIT 328 1 "x(0) = 0.2" "/-%\"xG6#\"\"!$\"\"#!\" \"" }{TEXT 329 1 " " }{TEXT -1 6 " and " }{TEXT 331 1 " " }{XPPEDIT 332 1 "D(x)(0) = 1" "/--%\"DG6#%\"xG6#\"\"!\"\"\"" }{TEXT 333 1 " " } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "a) Calculate and \+ Plot the Solution of the First Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart; with(DEtools): with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Name the differentia l equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "? := diff(x(t),t$2)+2*diff(x(t),t)+5*x(t) = cos(3*t );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "So lve the stated initial value problem." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "? := dsolve( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Name the \+ solution (but don't use " }{MPLTEXT 1 0 4 "x(t)" }{TEXT -1 3 " ):" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "? := unapply( ? , ? );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Plot the solution for " }{MPLTEXT 1 0 12 "t = \+ 0 .. 18" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot( ? , ? , color = plum, thickne ss = 3 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Name the plot:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "? := \":" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 57 "b) Calculate and Plot the Sol ution of the Second Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Name the differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "? := diff(x( t),t$2)+2*diff(x(t),t)+5*x(t) = cos(3*t) + 3*Dirac(t-5.2);;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Solve the state d initial value problem as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "? := map(z -> laplace(z, t, s), ? ) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute the two initial cond itions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "? := subs( \{ ? , ? \}, ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Solve for the Laplace transform:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " ? := laplace(x(t),t,s) = s olve(? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Apply the inverse Laplace transform to the right hand side:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "? := x(t) = invlaplace( ? , ? , ? ) ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "Plot this solution for " }{MPLTEXT 1 0 12 "t = 0 .. 18" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot( rhs(?) , ? , color = n avy, thickness = 1 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Name the plot:" }}{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 43 "Superimpose the plots of \+ parts (a) and (b)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 16 "Any \+ conclusions:" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyri ght and Author Information" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 45 "04 F00R4.mws A MapleV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 42 "Author: Brian E. Blank (3 November 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 tr ansfer, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permissi on 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 " W ashington 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 "7 \+ 10 11 0 0" 61 }{VIEWOPTS 1 1 0 3 4 1802 }