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-1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }2 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 473 47 "Wi1129-5: Stelsels diffe rentiaalvergelijkingen " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 0 "" }} {PARA 0 "" 0 "" {TEXT 257 105 "Vele technische en fysische problemen k unnen (bij benadering) gemodelleerd worden door stelsels lineaire " }} {PARA 0 "" 0 "" {TEXT 409 99 "differentiaalvergelijkingen van eerste o f hogere orde. Een inhomogeen stelsel lineaire eerste orde " }}{PARA 0 "" 0 "" {TEXT 464 60 "differentiaalvergelijkingen heeft de volgende \+ algemene vorm:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 333 1 " " }{TEXT 419 1 "x" }{TEXT 420 12 "'(t) = P(t) " }{TEXT 421 1 "x" }{TEXT 422 23 "(t) + g(t) (1)" }}{PARA 0 "" 0 "" {TEXT 258 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "met " }{TEXT 471 1 "x" } {TEXT 472 38 "(t) een vectorfunctie met componenten " }{XPPEDIT 18 0 " x[1](t):" "6#-&%\"xG6#\"\"\"6#%\"tG" }{TEXT 334 5 " t/m " }{XPPEDIT 18 0 "x[n](t):" "6#-&%\"xG6#%\"nG6#%\"tG" }{TEXT 335 38 " en P(t) een \+ nxn-matrix met elementen " }{XPPEDIT 18 0 "p[ij](t):" "6#-&%\"pG6#%#ij G6#%\"tG" }{TEXT 336 1 "." }}{PARA 0 "" 0 "" {TEXT 260 3 "Is " } {XPPEDIT 18 0 "g(t) =0:" "6#/-%\"gG6#%\"tG\"\"!" }{TEXT 337 30 " dan h eet het stelsel (1) een " }{TEXT 338 8 "homogeen" }{TEXT 339 13 " stel sel, is " }{XPPEDIT 18 0 "g(t)<>0:" "6#0-%\"gG6#%\"tG\"\"!" }{TEXT -1 0 "" }{TEXT 340 22 " dan heet het stelsel " }{TEXT 341 10 "inhomogeen " }{TEXT 342 2 ". " }}{PARA 0 "" 0 "" {TEXT 346 18 "Wanneer de matrix \+ " }{XPPEDIT 18 0 "P(t):" "6#-%\"PG6#%\"tG" }{TEXT 343 46 " niet van t \+ afhankelijk is spreken we van een " }{TEXT 344 31 " lineair stelsel me t konstante " }}{PARA 0 "" 0 "" {TEXT 347 13 "coefficienten" }{TEXT 345 26 ", meestal weergegeven door" }}{PARA 0 "" 0 "" {TEXT 261 0 "" } }{PARA 0 "" 0 "" {TEXT 262 1 " " }{TEXT 425 1 "x" }{TEXT 426 6 "' = A \+ " }{TEXT 423 1 "x" }{TEXT 424 9 " + g(t):" }}{PARA 0 "" 0 "" {TEXT 263 0 "" }}{PARA 0 "" 0 "" {TEXT 264 57 "Tot dit laatste type zullen w e ons voornamelijk beperken." }}{PARA 0 "" 0 "" {TEXT 265 0 "" }} {PARA 0 "" 0 "" {TEXT 266 101 "De theorie van lineaire stelsels vergt \+ enige kennis van lineaire algebra. We herhalen die hier niet. " }} {PARA 0 "" 0 "" {TEXT 348 74 "Voor enkele belangrijke Maple commando's verwijzen we naar de handleiding." }}{PARA 0 "" 0 "" {TEXT 267 0 "" } }{PARA 0 "" 0 "" {TEXT 268 0 "" }}{PARA 4 "" 0 "" {TEXT 269 44 "Stelse ls lineaire eerste orde vergelijkingen" }}{PARA 0 "" 0 "" {TEXT 270 0 "" }}{PARA 0 "" 0 "" {TEXT 271 9 "Voor een " }{TEXT 350 8 "homogeen" } {TEXT 351 58 " stelsel lineaire eerste orde differentiaalvergelijkinge n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 349 1 "x" }{TEXT 427 12 "'(t) = P(t) " }{TEXT 428 1 "x" }{TEXT 429 13 "(t) \+ (2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 36 " luidt de algemene oplossing: x(t) = " }{XPPEDIT 18 0 "c[1]*x[1](t):" " 6#*&&%\"cG6#\"\"\"F'-&%\"xG6#F'6#%\"tGF'" }{TEXT 352 9 " + ... + " } {XPPEDIT 18 0 "c[n]*x[n](t):" "6#*&&%\"cG6#%\"nG\"\"\"-&%\"xG6#F'6#%\" tGF(" }{TEXT 353 1 "," }}{PARA 0 "" 0 "" {TEXT 273 9 "waarbij " } {XPPEDIT 18 0 "x[1](t):" "6#-&%\"xG6#\"\"\"6#%\"tG" }{TEXT 354 5 " t/m " }{XPPEDIT 18 0 "x[n](t):" "6#-&%\"xG6#%\"nG6#%\"tG" }{TEXT 355 73 " een fundamenteel stelsel van oplossingen voor (2) vormt, dat wil zegg en " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x[1](t):" "6#-&%\"xG6#\"\"\"6#% \"tG" }{TEXT 356 5 " t/m " }{XPPEDIT 18 0 "x[n](t):" "6#-&%\"xG6#%\"nG 6#%\"tG" }{TEXT 357 63 " vormen een stelsel linear onafhankelijke opl ossingen van (2)." }}{PARA 0 "" 0 "" {TEXT 274 64 "We kunnen de algeme ne oplossing ook in matrixvorm schrijven als " }{XPPEDIT 18 0 "x(t) = \+ psi(t)*c:" "6#/-%\"xG6#%\"tG*&-%$psiG6#F'\"\"\"%\"cGF," }{TEXT 358 22 ", met c de vector met " }}{PARA 0 "" 0 "" {TEXT 359 10 "elementen " } {XPPEDIT 18 0 "c[1]:" "6#&%\"cG6#\"\"\"" }{TEXT 360 5 " t/m " } {XPPEDIT 18 0 "c[n]:" "6#&%\"cG6#%\"nG" }{TEXT 361 4 " en " }{XPPEDIT 18 0 "psi(t):" "6#-%$psiG6#%\"tG" }{TEXT 362 71 " een fundamentele mat rix, d.w.z. een matrix met als kolommen precies de" }}{PARA 0 "" 0 "" {TEXT 363 30 "n onafhankelijke oplossingen " }{XPPEDIT 18 0 "x[1](t): " "6#-&%\"xG6#\"\"\"6#%\"tG" }{TEXT 364 6 " t/m " }{XPPEDIT 18 0 "x[n ](t):" "6#-&%\"xG6#%\"nG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 275 0 "" }}{PARA 0 "" 0 "" {TEXT 276 88 "De Wronskiaan wordt wee r gebruikt om existentie en eenduidigheid van de oplossing aan te" }} {PARA 0 "" 0 "" {TEXT 277 48 "tonen. De Wronskiaan wordt nu gedefiniee rd als: " }{XPPEDIT 18 0 "W = det(psi(t)):" "6#/%\"WG-%$detG6#-%$psiG6 #%\"tG" }{TEXT 365 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):with(DEtools): " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 430 12 "voorbeeld :\n" }} {PARA 0 "" 0 "" {TEXT 279 20 "Gegeven het stelsel " }{TEXT 433 1 "x" } {TEXT 434 7 "'(t)=A " }{TEXT 435 1 "x" }{TEXT 436 10 "(t) met :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A := matrix(2,2,[[-2, 7], [1, 4]]); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 92 "We berekenen de eigenwaarden en de eigenvectoren van A met behulp \+ van wat Maple commando's :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigen vals(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 10 "opgave 1 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 431 46 "Geef nu de algemene oplossing van het stelsel." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 101 "Om het richtingsveld in het (x[1],x[2 ])-vlak te tekenen kunnen we het dfieldplot commando gebruiken :" }} {PARA 0 "" 0 "" {TEXT 432 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "d fieldplot(\{diff(x1(t),t)=-2*x1(t)+7*x2(t),diff(x2(t),t)=x1(t)+4*x2(t) \},\{x1(t),x2(t)\},t=0..1,x1=-7..7,x2=-7..7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 0 "" }} {PARA 0 "" 0 "" {TEXT 284 91 "Ook wanneer de matrix A complexe eigenwa arden heeft zoeken we een reeelwaardige oplossing. " }}{PARA 0 "" 0 " " {TEXT 366 91 "De oplossingen behorend bij eigenwaarden die als compl ex toegevoegde paren optreden kunnen " }}{PARA 0 "" 0 "" {TEXT 367 54 "in een reeel en een imaginair deel worden opgesplitst." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 0 "" }}{PARA 0 "" 0 "" {TEXT 286 9 "opgave 2 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):with(DEtools):" }}{PARA 0 "" 0 "" {TEXT 287 20 "Gegeven het stelsel " }{TEXT 446 1 "x" }{TEXT 447 9 "'(t) = A " } {TEXT 448 1 "x" }{TEXT 449 6 " met " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A := matrix(2,2,[[2, 1], [-17, -6]]);" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 51 "(a) Bereken de eigenwaar den en eigenvectoren van A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 289 96 "(b) Geef twee linear onafhankelij ke reeelwaardige oplossingen van het stelsel, en laat zien dat" }} {PARA 0 "" 0 "" {TEXT 290 52 "de oplossingen inderdaad lineair onafhan kelijk zijn." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Het fase- portret van het stelsel kan met behulp van het commando " }{TEXT 437 13 "phaseportrait" }{TEXT -1 18 " getekend worden :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart: with(linalg): with(DEtools):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 59 "deqns:=[D(x1)(t)=2*x1(t)+x2(t),D(x2)(t)=-17* x1(t)-6*x2(t)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars:=[x1(t),x2( t)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "inits:=[[x1(0)=1,x2(0)=0],[ x1(0)=-1,x2(0)=0],[x1(0)=0,x2(0)=2],[x1(0)=0,x2(0)=-2]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "phaseportrait(deqns,vars,t=0..50,inits,stepsi ze=.05);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 0 "" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 292 20 "Gegeven het stelsel " }{TEXT 438 1 "x" } {TEXT 439 9 "'(t) = A " }{TEXT 440 1 "x" }{TEXT 441 62 ". Een k-voudi ge wortel van het karakteristieke polynoom van A" }}{PARA 0 "" 0 "" {TEXT 293 25 "heet een eigenwaarde met " }{TEXT 368 16 "multipliciteit k" }{TEXT 369 47 ". Bij deze eigenwaarde kunnen k of minder dan k" }} {PARA 0 "" 0 "" {TEXT 294 94 "onafhankelijke eigenvectoren behoren. In het laatste geval moeten we op een andere manier dan " }}{PARA 0 "" 0 "" {TEXT 370 51 "gebruikelijk op zoek naar fundamentele oplossingen. " }}{PARA 0 "" 0 "" {TEXT 295 0 "" }}{PARA 0 "" 0 "" {TEXT 296 21 "Wan neer bijvoorbeeld " }{XPPEDIT 18 0 "rho:" "6#%$rhoG" }{TEXT 371 56 " e en dubbele eigenwaarde is met slechts een eigenvector " }{XPPEDIT 18 0 "v[1]:" "6#&%\"vG6#\"\"\"" }{TEXT 372 11 " dan hebben" }}{PARA 0 "" 0 "" {TEXT 373 17 "we \351\351n oplossing " }{XPPEDIT 18 0 "x[1](t)=v[ 1]*exp(rho*t):" "6#/-&%\"xG6#\"\"\"6#%\"tG*&&%\"vG6#F(F(-%$expG6#*&%$r hoGF(F*F(F(" }{TEXT 374 38 ". Een tweede onafhankelijke oplossing " } {XPPEDIT 18 0 "x[2](t):" "6#-&%\"xG6#\"\"#6#%\"tG" }{TEXT 375 10 " zoe ken we" }}{PARA 0 "" 0 "" {TEXT 297 11 "in de vorm " }{XPPEDIT 18 0 "x [2](t)=v[1]*t*exp(rho*t) + v[2]*exp(rho*t):" "6#/-&%\"xG6#\"\"#6#%\"tG ,&*(&%\"vG6#\"\"\"F0F*F0-%$expG6#*&%$rhoGF0F*F0F0F0*&&F.6#F(F0-F26#*&F 5F0F*F0F0F0" }{TEXT 376 9 " waarbij " }{XPPEDIT 18 0 "v[2]:" "6#&%\"vG 6#\"\"#" }{TEXT 377 33 " een zogenaamde gegeneraliseerde " }}{PARA 0 " " 0 "" {TEXT 378 45 "eigenwaarde is die volgt uit de vergelijking " } {XPPEDIT 18 0 "(A-rho*I)*v[2] = v[1]:" "6#/*&,&%\"AG\"\"\"*&%$rhoGF'% \"IGF'!\"\"F'&%\"vG6#\"\"#F'&F-6#F'" }{TEXT 379 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 0 "" }}{PARA 0 "" 0 "" {TEXT 299 9 "opgave 3:" }} {PARA 0 "" 0 "" {TEXT 300 0 "" }}{PARA 0 "" 0 "" {TEXT 301 20 "Gegeven het stelsel " }{TEXT 444 1 "x" }{TEXT 445 9 "'(t) = A " }{TEXT 442 1 "x" }{TEXT 443 6 " met :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A:=matrix(2,2,[[1,25],[-1,-9]]);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 380 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 28 "(a) Bereken de eigenwaarden " }{XPPEDIT 18 0 "r[i]:" "6# &%\"rG6#%\"iG" }{TEXT 381 19 " en de eigenvector " }{XPPEDIT 18 0 "v[1 ]:" "6#&%\"vG6#\"\"\"" }{TEXT 382 7 " van A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 303 49 "(b) Bereken \+ de gegeneraliseerde eigenvector v[2]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 304 47 "(c) Geef de algemene op lossing van het stelsel." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 305 44 "(d) Teken het richtingsveld van h et stelsel." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 93 "(e) De algemene oplossing van het stelsel word hie ronder berekend. Is deze oplossing correct?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "dvs:=\{diff(x1(t),t)=x1(t)+25*x2(t),diff(x2(t),t)= -x1(t)-9*x2(t)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(dvs,\{x 1(t),x2(t)\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 306 0 "" }}{PARA 0 "" 0 "" {TEXT 307 36 "Een speciale fundamentele matrix is " }{XPPEDIT 18 0 "Phi(t):" "6#-%$PhiG6#%\"tG" }{TEXT 383 32 ", opgebouwd uit de oplos singen " }{XPPEDIT 18 0 "x[1](t):" "6#-&%\"xG6#\"\"\"6#%\"tG" }{TEXT 386 6 " t/m/ " }{XPPEDIT 18 0 "x[n](t):" "6#-&%\"xG6#%\"nG6#%\"tG" } {TEXT 384 4 " van" }}{PARA 0 "" 0 "" {TEXT 385 54 "x'(t) = P(t) x(t) d ie aan de speciale beginvoorwaarde " }{XPPEDIT 18 0 "x[j](t[0])=e[j]: " "6#/-&%\"xG6#%\"jG6#&%\"tG6#\"\"!&%\"eG6#F(" }{TEXT 387 20 " voldoen . Hierin is " }{XPPEDIT 18 0 "e[j]:" "6#&%\"eG6#%\"jG" }{TEXT 388 10 " de j-de " }}{PARA 0 "" 0 "" {TEXT 463 16 "eenheidsvector. " } {XPPEDIT 18 0 "Phi(t):" "6#-%$PhiG6#%\"tG" }{TEXT 389 29 " heeft dus d e eigenschap dat " }{XPPEDIT 18 0 "Phi(t[0])=I:" "6#/-%$PhiG6#&%\"tG6# \"\"!%\"IG" }{TEXT 390 2 ". " }}{PARA 0 "" 0 "" {TEXT 308 17 "Voor het stelsel " }{TEXT 459 1 "x" }{TEXT 460 9 "'(t) = A " }{TEXT 461 1 "x" }{TEXT 462 19 " met beginconditie " }{XPPEDIT 18 0 "x(0)=x[0]:" "6#/-% \"xG6#\"\"!&F%6#F'" }{TEXT 411 1 " " }{TEXT -1 8 " geldt " }{XPPEDIT 18 0 "Phi(t) = exp(A*t):" "6#/-%$PhiG6#%\"tG-%$expG6#*&%\"AG\"\"\"F'F- " }{TEXT -1 43 ". De oplossing van het beginwaardeprobleem " }{TEXT 412 11 "wordt dan: " }{XPPEDIT 18 0 "x(t) = exp(A*t)*x[0]:" "6#/-%\"xG 6#%\"tG*&-%$expG6#*&%\"AG\"\"\"F'F.F.&F%6#\"\"!F." }{TEXT -1 3 " . " } }{PARA 0 "" 0 "" {TEXT 410 23 "De fundamentele matrix " }{XPPEDIT 18 0 "exp(A*t):" "6#-%$expG6#*&%\"AG\"\"\"%\"tGF(" }{TEXT 413 47 " kan in Maple in \351\351n stap berekend worden met " }{TEXT 391 16 "exponen tial(A,t)" }{TEXT 392 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 414 4 "N.B." }{TEXT -1 75 " Matrixvermenigvuldiging van \+ de matrix A met de vector b gaat in Maple met " }{TEXT 416 11 "evalm(A &*b)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 309 0 "" }} {PARA 0 "" 0 "" {TEXT 310 9 "opgave 4:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):with(DEtoo ls):" }}{PARA 0 "" 0 "" {TEXT 311 0 "" }}{PARA 0 "" 0 "" {TEXT 312 19 "Gegeven de matrix :" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A:=matrix(2,2,[[1,1],[0,2]]);" }}{PARA 0 "" 0 "" {TEXT 313 12 "(a) Bereken " }{XPPEDIT 18 0 "exp(A*t):" "6#-%$expG6#*&%\"AG\"\"\"%\"tGF( " }{TEXT 393 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 50 "(b) Geef de oplossing van het beginwaard eprobleem " }{TEXT 450 1 "x" }{TEXT 451 9 "'(t) = A " }{TEXT 452 1 "x " }{TEXT 453 4 " met" }}{PARA 0 "" 0 "" {TEXT 315 16 "beginvoorwaarde \+ " }{XPPEDIT 18 0 "x(0)=(1,1):" "6#/-%\"xG6#\"\"!6$\"\"\"F)" }{TEXT 394 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 0 "" }}{PARA 3 "" 0 "" {TEXT 317 28 "Inhomogene lineair e stelsels" }}{PARA 0 "" 0 "" {TEXT 318 0 "" }}{PARA 0 "" 0 "" {TEXT 319 84 "Tot slot gaan we nog in op enkele aspecten van het inhomogene \+ stelsel vergelijkingen" }}{PARA 0 "" 0 "" {TEXT 320 1 "x" }{TEXT 454 9 "'(t) = A " }{TEXT 455 1 "x" }{TEXT 456 12 " + g(t), " }{XPPEDIT 18 0 "x(0)=x[0]:" "6#/-%\"xG6#\"\"!&F%6#F'" }{TEXT 395 10 ". (1) " }}{PARA 0 "" 0 "" {TEXT 321 72 "Analoog aan de enkelvoudige differen tiaalvergelijking kan de oplossing " }{TEXT 457 1 "x" }{TEXT 458 15 " (t) geschreven " }}{PARA 0 "" 0 "" {TEXT 396 64 "worden als de som van de oplossing van de homogene vergelijking " }{XPPEDIT 18 0 "x[h](t): " "6#-&%\"xG6#%\"hG6#%\"tG" }{TEXT 397 21 " en een partikuliere " }} {PARA 0 "" 0 "" {TEXT 398 10 "oplossing " }{XPPEDIT 18 0 "x[p](t):" "6 #-&%\"xG6#%\"pG6#%\"tG" }{TEXT 399 32 " van de inhomogene vergelijking ." }}{PARA 0 "" 0 "" {TEXT 322 54 "De algemene oplossing van de homoge ne vergelijking is " }{XPPEDIT 18 0 "x[h](t) = exp(A*t)*x[0]:" "6#/-&% \"xG6#%\"hG6#%\"tG*&-%$expG6#*&%\"AG\"\"\"F*F1F1&F&6#\"\"!F1" }{TEXT 400 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 0 "" }}{PARA 0 "" 0 "" {TEXT 324 9 "opgave 5:" }}{PARA 0 "" 0 "" {TEXT 325 41 "Deze opgave mo et U met de hand uitwerken." }}{PARA 0 "" 0 "" {TEXT 326 98 "Om de opl ossing van de inhomogene vergelijking te vinden veronderstellen we nu \+ dat de particuliere" }}{PARA 0 "" 0 "" {TEXT 401 31 "oplossing te schr ijven is als " }{XPPEDIT 18 0 "x[p](t) = exp(A*t)*u(t):" "6#/-&%\"xG6 #%\"pG6#%\"tG*&-%$expG6#*&%\"AG\"\"\"F*F1F1-%\"uG6#F*F1" }{TEXT 402 20 ", dat wil zeggen we " }{TEXT 406 21 "vari\353ren de constante" } {TEXT 407 17 " uit de homogene " }}{PARA 0 "" 0 "" {TEXT 465 10 "oplos sing." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 327 44 "(a) Leidt de differenti aalvergelijking voor " }{XPPEDIT 18 0 "u(t):" "6#-%\"uG6#%\"tG" } {TEXT 403 29 " af en geef de oplossing van " }{XPPEDIT 18 0 "u(t):" "6 #-%\"uG6#%\"tG" }{TEXT 404 20 " in de vorm van een " }}{PARA 0 "" 0 " " {TEXT 405 10 "integraal." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 328 89 "(b) Laat zien dat de particulier e oplossing van (1) in integraalvorm gegeven wordt door:" }}{PARA 0 " " 0 "" {XPPEDIT 18 0 "x(t) = int(exp(A*(t-s))*g(s),s = 0 .. t):" "6#/- %\"xG6#%\"tG-%$intG6$*&-%$expG6#*&%\"AG\"\"\",&F'F1%\"sG!\"\"F1F1-%\"g G6#F3F1/F3;\"\"!F'" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "(c) Geef nu de algemene oplo ssing van (1)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 329 0 "" }}{PARA 0 "" 0 "" {TEXT 330 9 "opgave 6:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):with(DEtools): " }{TEXT 331 0 "" }}{PARA 0 "" 0 "" {TEXT 332 56 "We passen het gevond ene uit opgave 6 toe op het stelsel " }{TEXT 466 1 "x" }{TEXT 467 9 "' (t) = A " }{TEXT 468 1 "x" }{TEXT 469 25 " + g(t) met A de matrix :" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A: =matrix(2,2,[[2,-5],[4,-2]]);" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 408 22 " en g(t) gegeven door " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g := t -> vector([cos(t), 0]):" }}{PARA 0 "" 0 "" {TEXT -1 60 "W e beginnen met het berekenen van de particuliere oplossing." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(a) Bereken " }{XPPEDIT 18 0 "exp (A*(t-s)):" "6#-%$expG6#*&%\"AG\"\"\",&%\"tGF(%\"sG!\"\"F(" }{TEXT -1 26 " en noem het resultaat B." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B :=exponential(A,t-s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "(b) Bere ken " }{XPPEDIT 18 0 "exp(A*t-s)*g(t):" "6#*&-%$expG6#,&*&%\"AG\"\"\"% \"tGF*F*%\"sG!\"\"F*-%\"gG6#F+F*" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "Bmaalgs:=evalm(B&*g(s));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 12 "(c) Bereken " }{XPPEDIT 18 0 "int(exp(A*(t-s))*g(s),s = 0 .. t):" "6#-%$intG6$*&-%$expG6#*&%\"AG\"\"\",&%\"tGF,%\"sG!\"\"F,F, -%\"gG6#F/F,/F/;\"\"!F." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xp:=map(int,Bmaalgs,s=0..t);" }}{PARA 0 "" 0 "" {TEXT -1 19 "( Door het commando " }{TEXT 415 3 "map" }{TEXT -1 32 " te gebruiken wor dt de opdracht " }{TEXT 417 3 "int" }{TEXT -1 22 " op ieder element va n " }{XPPEDIT 18 0 "exp(A*t-s)" "6#-%$expG6#,&*&%\"AG\"\"\"%\"tGF)F)% \"sG!\"\"" }{TEXT -1 13 " uitgevoerd)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "(d) Wat wordt nu de oplossing van het stelsel bij beginco nditie " }{XPPEDIT 18 0 "x[0]=(1,1):" "6#/&%\"xG6#\"\"!6$\"\"\"F)" } {TEXT -1 2 " ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 418 20 "TH & KL, May-July 99" }}{PARA 257 "" 0 "" {TEXT 470 21 "Modified May 2002, KL" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }