{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 "" -1 259 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "Courier" 1 12 255 0 0 1 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 255 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 18 0 0 255 1 0 0 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 "Cou rier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 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 "" 1 18 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 18 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 296 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 304 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "Courier" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 1 12 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 1 12 0 128 0 1 1 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 314 "" 1 12 255 0 255 1 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 1 12 255 0 255 1 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 0 1 0 128 0 1 1 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 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 320 "" 0 1 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 321 "" 0 1 128 0 128 1 1 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "Courier " 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 331 "Courier" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 332 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 335 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 336 "" 1 18 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 337 "" 0 1 255 0 0 1 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 }{CSTYLE "" -1 339 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 340 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 341 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 342 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 343 "Courier" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 344 "" 0 1 0 128 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 345 "" 1 14 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 346 "Courier" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 347 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 348 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 349 "" 1 14 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 350 "Courier " 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 351 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 352 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 353 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 354 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 355 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 356 "" 1 14 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 357 "Cour ier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 358 "Courier" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 359 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 360 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 361 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 366 "" 1 18 255 0 255 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 367 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 1 18 255 0 255 1 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 1 18 255 0 255 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 370 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 371 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 372 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 373 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 375 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 376 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 1 12 0 128 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 378 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 380 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 383 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 19 384 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 387 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 388 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 389 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 390 "Couri er" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 391 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 392 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 393 "" 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 395 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 396 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 397 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 398 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 399 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 400 "Cou rier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 401 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 403 "" 1 12 0 128 0 1 1 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 Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Tit le" 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 "" 0 270 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 "" 0 271 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 279 16 "Direction Fields" }} {PARA 258 "" 0 "" {TEXT 256 4 "HW 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expan d a section. Click on a [-] sign to collapse a section. To do these ex ercises you will have to insert execution groups. That can be done by \+ clicking on the toolbar icon that looks like \"[>\". It can also be do ne via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 271 " " 0 "" {TEXT 366 73 "If you do your work as part of a team, then fill \+ in both lines below and " }{TEXT 368 32 "hand in only one report per t eam" }{TEXT 369 36 ". Otherwise, delete the second line." }{TEXT 367 1 "\n" }}{PARA 0 "" 0 "" {TEXT 364 20 "Student Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 365 20 "Student Name a nd ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " } {TEXT 257 5 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "wor ksheet" "" }{TEXT -1 94 ", you will be asked to plot direction fields of first order differential equations. The main" }{TEXT 310 7 " MAP LE" }{TEXT -1 36 " functions that can be used are " }{TEXT 400 4 " diff" }{TEXT -1 3 ", " }{TEXT 401 6 "DEplot" }{TEXT -1 3 ", " } {TEXT 399 6 "dsolve" }{TEXT -1 4 ", " }{TEXT 311 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 0 {PARA 3 "" 0 "" {TEXT -1 16 "Reports (Read!)\n" }} {PARA 0 "" 0 "" {TEXT -1 206 "There are two main themes to this sectio n: (1) reports should be sensibly prepared, and (2) the resources used in preparing a report should be considered. This section is written w ith regard to the use of " }{TEXT 377 5 "MAPLE" }{TEXT 378 1 " " } {TEXT -1 77 " but if you use a different package, then you should use \+ the same discretion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 370 30 "Reports that you prepare with " }{TEXT 312 5 "MAPLE" } {TEXT 371 112 " should be prepared with the same care that you would devote to laboratory reports in biology and chemistry. " }{TEXT 314 46 "A report should not be a diary or history of a" }{TEXT 372 1 " " } {TEXT 313 1 " " }{TEXT 403 5 "MAPLE" }{TEXT 373 2 " " }{TEXT 315 52 " session. Delete what is not needed for the report." }{TEXT 374 25 " \+ All lines of the form " }{TEXT 280 6 "?topic" }{TEXT 375 80 " (that arise from help queries) should be erased. All errors should be eras ed." }{TEXT 376 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 317 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper resou rces that you are using. All commands must be terminated - either with the standard terminator, the semicolon, or the " }{TEXT 321 17 "silen t terminator" }{TEXT -1 57 ", the colon. When you assign a variable, f or example \n \"" }{TEXT 359 7 "x := 5;" }{TEXT -1 30 " \", there is no need to have " }{TEXT 318 5 "MAPLE" }{TEXT -1 12 " echo back " } {TEXT 281 6 "x := 5" }{TEXT -1 89 ". When this is printed, it simply \+ wastes paper and ink. Choose the silent terminator \"" }{TEXT 282 7 "x := 5:" }{TEXT -1 70 " \" instead. When you load a package (withou t the silent terminator)," }{TEXT 283 1 " " }{TEXT -1 2 " " }{TEXT 319 5 "MAPLE" }{TEXT -1 123 " will list the commands that become avail able with the package. This is fine - it will help you become familiar with what " }{TEXT 320 5 "MAPLE" }{TEXT -1 125 " makes available. Ho wever, these commands should not be part of a lab report. Reload the p ackage with the silent terminator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 79 "Much of the text in this worksheet shoul d be deleted. For example, delete the " }{TEXT 287 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 288 8 "Keywords" }{TEXT -1 34 " sections. D elete this section on " }{TEXT 289 7 "Reports" }{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 writ ten. In particular, remember that the ditto refers to the result of t he last executed command - not the result of the command that physical ly 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 "DEplot" "" }{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 "disp lay" 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 "p lot,options" 2 "plot,options" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "res tart" 2 "restart" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "simplify" 2 "si mplify" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "testeq" 2 "testeq" "" }{TEXT -1 1 "." }}} {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 322 18 "Backgroud Reading:" }{TEXT -1 21 " The s ection titled \"" }{TEXT 323 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 5 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Verify that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{TEXT 379 8 " \+ " }{XPPEDIT 380 1 "y(x)=A*exp(-2*a/sqrt(x))*(x^(3/4)+x^(5/4)/2/ a)+B*exp(2*a/sqrt(x))*(x^(3/4)-x^(5/4)/2/a)" "/-%\"yG6#%\"xG,&*(%\"AG \"\"\"-%$expG6#,$*(\"\"#F*%\"aGF*-%%sqrtG6#F&!\"\"F5F*,&)F&*&\"\"$F*\" \"%F5F**()F&*&\"\"&F*F:F5F*F0F5F1F5F*F*F**(%\"BGF*-F,6#*(F0F*F1F*-F36# F&F5F*,&)F&*&F9F*F:F5F**()F&*&F>F*F:F5F*F0F5F1F5F5F*F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 25 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "is a solution of the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 295 27 " " }{XPPEDIT 296 1 "di ff( y(x), x,x) = (a^2+5/16*x)*y(x)/x^3" "/-%%diffG6%-%\"yG6#%\"xGF)F)* (,&*$%\"aG\"\"#\"\"\"*(\"\"&F/\"#;!\"\"F)F/F/F/-F'6#F)F/*$F)\"\"$F3" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 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 14 "subs( ? , ? );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 45 "Exercise 2 Tr ajectories and Direction Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 324 18 "Backgroud Reading:" }{TEXT -1 14 " Work sheet " }{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[dfiel dplot]" "" }{TEXT -1 8 " and " }{HYPERLNK 17 "display" 2 "plots[dis play]" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 145 "If a 1000 kg bo mb is subject only to the force of the Earth's gravity and a quadratic drag law, then the differential equation for its velocity " }{TEXT 298 1 "v" }{TEXT 301 1 " " }{TEXT -1 14 " (measured in " }{TEXT 297 3 "m/s" }{TEXT -1 25 ") is\n\n " }{TEXT 383 12 " \+ " }{XPPEDIT 384 1 "diff(v(t),t) = -g+.2e-4*v(t)^2" "/-%%diffG6$- %\"vG6#%\"tGF),&%\"gG!\"\"*&$\"\"#!\"&\"\"\"*$-F'6#F)F/F1F1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 8 "Here " }{TEXT 303 1 "g" }{TEXT 307 1 " " }{TEXT 304 6 "= 9.81" }{TEXT 302 89 " m/s/s. (The coefficient 0.00002 depends on various factors which are not relevant here.)" }}{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 290 27 "[ 0 , 30 ] x [ -300 , 100 ]" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 122 "Do this by replacing the question marks \+ with appropriate arguments. For the last execution group, look up the available " }{HYPERLNK 17 "color choices" 2 "color" "" }{TEXT -1 105 " and pick one you like. (Light colors do not alwayys print out \+ well.) Execute each line in succession." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with(plots ): with(DEtools): # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ode := diff(v(t), t) = -9.81 +.2e-4*v(t)^2 ; # Execute!" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "dfieldplot( ? , ? , t = 0 .. 30, v = ? , 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 pl ot 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 0 {PARA 4 "" 0 "" {TEXT -1 53 " b) Plotting the Solution of One Initial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 15 "If the bomb is " }{TEXT 387 7 "dropped" }{TEXT -1 1 " \+ " }{TEXT 260 1 "," }{TEXT -1 32 " find an explicit formula for " } {TEXT 275 4 "v(t)" }{TEXT -1 5 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "soln1 := dsolve( \{ ? , v(0)= ? \} , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Define the solu tion of this equation to be " }{TEXT 391 2 "v1" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v1 := unapply( rhs(?) , ? ); " }}}{PARA 0 "" 0 "" {TEXT -1 10 "\n\nPlot " }{TEXT 390 5 "v1(t)" } {TEXT -1 8 " for " }{TEXT 389 1 "t" }{TEXT -1 20 " in the interva l " }{TEXT 388 10 "[ 0 , 30 ]" }{TEXT -1 25 ". Use the plot option \+ " }{TEXT 392 13 "thickness = 2" }{TEXT -1 9 ". (See " }{HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 4 ". )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(v1(t), t = 0 .. 30, 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := \" :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 58 "c) Plotting the Solution of a Second In itial Value Problem" }}{PARA 0 "" 0 "" {TEXT -1 50 "If a missile is pr opelled with initial velocity " }{TEXT 266 10 "v(0) = 100" }{TEXT 329 1 " " }{TEXT 265 6 "m/s, " }{TEXT -1 30 "find an explicit formula for " }{TEXT 277 4 "v(t)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "? := dsolve( \{ ? , ? \} , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Define the solution of this equation to be " }{TEXT 396 2 "v2 " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "v2 := unappl y( ? , ? );" }}}{PARA 0 "" 0 "" {TEXT -1 10 "\n\nPlot " }{TEXT 395 5 "v2(t)" }{TEXT -1 8 " for " }{TEXT 398 1 "t" }{TEXT -1 19 " in t he interval " }{TEXT 393 10 "[ 0 , 30 ]" }{TEXT -1 25 ". Use the pl ot option " }{TEXT 397 13 "thickness = 2" }{TEXT -1 9 ". (See " } {HYPERLNK 17 "plot,options" 2 "plot,options" "" }{TEXT -1 4 ". )\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(?, ?, thickness = 2, co lor = ? );" }}}{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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "? := \" :" }}}{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 27 "d) Superimposi ng Everything" }}{PARA 0 "" 0 "" {TEXT -1 51 "Superimpose the plots t hat 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 c ommand " }{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 23 "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 362 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 345 5 "MAPLE" }{TEXT -1 22 " kernel and load the " }{TEXT 346 7 "DEtools" }{TEXT -1 68 " package . (Place the cursor anywhere in the next line and press the " }{TEXT 347 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 348 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 350 10 "dfieldplot" }{TEXT -1 76 " to plot the direction field of the given equation in the viewing win dow " }{TEXT 351 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 352 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 353 1 "x" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 36 "What can you conclude the value of " }{TEXT 354 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 355 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 356 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 349 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 360 10 " " }{XPPEDIT 361 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 344 5 " MAPLE" }{TEXT -1 12 " Function " }{TEXT 343 6 "DEplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 363 18 "Backgroud Reading: " }{TEXT -1 21 " The help page for " }{HYPERLNK 17 "DEplot" 2 "DEtoo ls[DEplot]" "" }{TEXT -1 103 " is rather involved. The next execution group provides you with an example of how to use the function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Look at the commands in the next execution grou p 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 284 4 " " }{TEXT 330 46 " y'(x) = x^3/(1+x^4) + 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 read 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 0 139 "restart: with(DEtools): wi th(plots):\node := diff(y(x), x) = x^3/(1+x^4) + sin(y(x)): \nDEplot(o de , [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 above the worksheet, to the left of the p lot 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 "Notic e that the graph of the solution becomes rather flat as " }{TEXT 333 1 "x" }{TEXT -1 15 " increases to " }{TEXT 331 3 " 10" }{TEXT -1 8 ". Use " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }{TEXT -1 42 " to stu dy the behavior for values of " }{TEXT 332 1 "x" }{TEXT -1 9 " betwe en " }{MPLTEXT 1 0 3 "150" }{TEXT -1 6 " and " }{MPLTEXT 1 0 3 "200" }{TEXT -1 44 ". Describe what appears to be happening to" }{TEXT 335 5 " y(x)" }{TEXT -1 6 " as " }{TEXT 334 1 "x" }{TEXT -1 20 " ten ds 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 Proble ms" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use " }{HYPERLNK 17 "DEplot" 2 "DEp lot" "" }{TEXT -1 49 " to study solutions to the Initial Value Proble m" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 46 "y'( x) = x^3/(1+x^4) + 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 286 47 "y '(x) = x^3/(1+x^4) + sin(y(x)), y(-5) = - 15" }{TEXT 336 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{TEXT 342 1 "x" }{TEXT -1 11 " between \+ " }{TEXT 338 2 "-5" }{TEXT -1 7 " and " }{TEXT 337 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 are t he possible" }}{PARA 0 "" 0 "" {TEXT -1 9 "values of" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ " }{TEXT 339 4 " " }{XPPEDIT 340 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 341 4 "y(x)" }{TEXT -1 191 " satisfies the given equation? (Click on the curves to guess this limit. For the first you should be able to see what exact number the \+ limit is. Use that value as a hint for the second.)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 100 "c) Using a Differential Equ ation to Understand Solution Curves (without having an explicit formul a) " }}{PARA 0 "" 0 "" {TEXT -1 74 " Use the differential equation to \+ explain the reason for the behavior of " }{TEXT 358 4 "y(x)" }{TEXT -1 6 " as " }{TEXT 357 1 "x" }{TEXT -1 73 " tends to infinity. (You should aim for plausibility rather than rigor.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information " }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 45 "01S01R4.mws A MapleV Rel ease 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 43 "Author: Brian E. Blank (01 February 2001)" }}{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 permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For more info rmation, 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. Lo uis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 63130" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " T elephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " \+ e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 " " }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 2001 Brian E. Blan k, All Rights Reserved." }}}}}{MARK "3 1 3" 77 }{VIEWOPTS 1 1 0 3 4 1802 }