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{CSTYLE "" -1 -1 "He lvetica" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 336 25 "Moderne Physik mit Mapl e " }}{PARA 259 "" 0 "" {TEXT 337 9 "PDF-Buch " }{URLLINK 17 "Moderne \+ Physik mit Maple" 4 "http://mikomma.de/fh/modphys.pdf" "" }}{PARA 259 "" 0 "" {TEXT -1 19 "Update auf Maple 10" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 13 "Kapitel A.2.1" }}{PARA 259 " " 0 "" {TEXT -1 23 "Worksheet mld1g1_10.mws" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "DG erster Ordnung" {TEXT -1 24 "Dr. M. Komma \+ " }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 42 "DG 1. Ordnung mit konstanten Koeffizienten" }}{PARA 19 "" 0 "" {TEXT 256 13 "Ein \334berblick" }} {PARA 258 "" 0 "" {TEXT -1 130 "(Aus 'Moderne Physik mit Maple' Komma , International Thomson Publishing Bonn 1995 filename : mld1g1.ms 24.7.94)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1272 "Die linearen Differentialgleichungen mi t konstanten Koeffizienten geh\366ren wohl zum wichtigsten Gleichungst yp der Physik. Sie erscheinen in weiten Bereichen als der Prototyp der Bewegungsgleichung einer deterministisch aufgefa\337ten Natur. Das re icht vom Zerfall und vom Wachstum \374ber periodische Vorg\344nge, abe r auch Diffusionsprozesse bis in die Quantenphysik. Es scheint einfach nat\374rlich zu sein, da\337 zwischen der \304nderung einer Gr\366 \337e und ihrem Momentanwert Proportionalit\344ten bestehen. Solche Zu sammenh\344nge werden durch die Exponentialfunktion beschrieben aber n ur dann in voller Allgemeinheit, wenn man komplexe Argumente zul\344 \337t. Es ist das Anliegen dieser Kapitel, diese Zusammenh\344nge zu b eleuchten. Wie schon im vorangehenden Abschnitt wird nicht beabsichtig t, eine komplette Theorie darzustellen, sondern die M\366glichkeiten e ines CAS auszun\374tzen, um einen kleinen Streifzug durch die Welt der Exponentialfunktion zu machen. Die dabei gekn\374pften Assoziationen \+ reichen von der L\366sung der DGn \374ber Diffrenzengleichungen und di e geometrische Reihe bis hin zu Eigenwerten und Eigenvektoren. Sie k \366nnen vom (inter-)aktiven Leser weiter verfolgt werden, indem er Pa rameter \344ndert, Darstellungsm\366glichkeiten auslotet, einzelne Tei le der worksheets umordnet und neu kombiniert." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Weil wir es mit \"Bewegu ngen\" zu tun haben, w\344hlen wir die Zeit t als unabh\344ngige Varia ble (und den \"Ort\" x als abh\344ngige Variable)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "Der Standard-Zugang zur \+ linearen DG (mit konstanten Koeffizienten wird im Folgenden weggelasse n) sieht gew\366hnlich so aus. Die homogene DG " }{TEXT 257 3 "ay' " }{TEXT -1 3 " + " }{TEXT 258 2 "by" }{TEXT -1 42 " = 0 kann auch geschrieben werden als:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart; with(DEtools): with(plots) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dg:=diff(x(t),t)+k*x(t )=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dgG/,&-%%diffG6$-%\"xG6#%\" tGF-\"\"\"*&%\"kGF.F*F.F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(dg,x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#% \"tG*&%$_C1G\"\"\"-%$expG6#,$*&%\"kGF*F'F*!\"\"F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "Zerfall, exponentie ller" {TEXT -1 348 "Je nach dem Vorzeichen von k liegt also exponentie ller Zerfall oder exponentielles Wachstum vor. Das ist allgemein bekan nt und man ben\366tigt daf\374r kein CAS. Aber schon wenn man etwa den Zerfall einer radioaktiven Substanz eine Generation weiter verfolgen \+ will, hat man wohl mit den herk\366mmlichen Methoden die L\366sung nic ht so schnell parat wie Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dgln:=diff(A(t), t)=-ka*A(t)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " diff(B(t),t)= ka*A(t)-kb*B(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dglnG6$/-%%dif fG6$-%\"AG6#%\"tGF-,$*&%#kaG\"\"\"F*F1!\"\"/-F(6$-%\"BGF,F-,&F/F1*&%#k bGF1F6F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sol:=dsolve( \{dgln\},\{A(t),B(t)\},method=laplace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/-%\"AG6#%\"tG*&-F(6#\"\"!\"\"\"-%$expG6#,$*&%#kaGF/F*F /!\"\"F//-%\"BGF),&*&-F16#,$*&%#kbGF/F*F/F6F/-F9F-F/F/**,&F " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "phaseportrait()" {TEXT -1 15 "Mit dem Befehl " } {TEXT 259 10 "DEplot3d()" }{TEXT -1 270 " (Package DEtools) kann man s ich einen schnellen \334berblick \374ber die L\366sung verschaffen. Ma n erh\344lt eine 3D-Darstellung, die man mit der Maus so drehen kann, \+ da\337 A(t), B(t) oder B(A) zweidimensional dargestellt werden. Dabei \+ ist B(A) das Phasenportrait, das man auch mit " }{TEXT 260 8 "DEplot() " }{TEXT -1 6 " oder " }{TEXT 261 16 "phaseportrait() " }{TEXT -1 126 "zusammen mit dem Richtungsfeld erh\344lt. 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Mu\337 das immer so sein? 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Und ganz nebenbei l\344\337t sich gleich auch noc h die Frage kl\344ren, was f\374r " }{TEXT 264 2 "kb" }{TEXT -1 3 " = \+ " }{TEXT 265 2 "ka" }{TEXT -1 10 " passiert:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit(%,kb=ka);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&%\"AG\"\"\"-%\"BG6#\"\"!F'F'*(F&F'-%#lnG6#*&F&F'-F&F*!\"\"F'F 1F'F2F'F1F2" }}}{EXCHG {PARA 0 "" 0 "Radioaktivitaet " {TEXT -1 66 "Pl ot der Aktivit\344ten f\374r verschiedene Werte der Zerfallskonstante \+ " }{TEXT 266 2 "kb" }{TEXT -1 18 " (spielen Sie mit " }{TEXT 267 2 "ka " }{TEXT -1 25 " und dem Laufbereich von " }{TEXT 268 2 "kb" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ka:=1: kb:='kb' : A(0):=2: B(0):=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot (\{ka*A(t),seq(eval(kb*B(t)),kb=seq(0.3*ii,ii=1..9))\},t=0..10);" }} {PARA 13 "" 1 "" {GLPLOT2D 510 510 510 {PLOTDATA 2 "6.-%'CURVESG6$7Z7$ $\"\"!F)$\"\"#F)7$$\"+S&)G\\a!#6$\"+#\\IR*=!\"*7$$\"+3x&)*3\"!#5$\"+bj 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1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "Gleichgewicht, radioaktiver \+ Zerfall" {TEXT -1 75 "Man sieht so auf einen Blick die verschiedenen G leichgewichtstypen (ideal: " }{TEXT 273 6 "dB/dt " }{TEXT -1 46 "= 0 \+ - vgl. die zweite DG - , fortschreitend: " }{TEXT 274 2 "kb" }{TEXT -1 3 " > " }{TEXT 275 2 "ka" }{TEXT -1 13 " und sekul\344r " }{TEXT 276 2 "kb" }{TEXT -1 4 " >> " }{TEXT 277 2 "ka" }{TEXT -1 374 "). Und \+ nat\374rlich l\344\337t sich - einmal mehr - die Fragestellung leicht \+ umkehren und beanworten: F\374r welche Wertepaare der Zerfallskonstant en bekommt man zu einem vorgegebenem Zeitpunkt die gleiche Gesamtaktiv it\344t und wann ist sie am gr\366\337ten? Wie h\344ngt das Maximum vo m Zeitpunkt ab? (Wenn man die Zeit variiert findet Maple ein paar unre alistische Inseln, die sich aber mit der " }{TEXT 278 4 "grid" }{TEXT -1 149 "-option ausb\374geln lassen. Trick: Im Argument von A und B mu \337 't' geschrieben werden, damit nicht die Funktionen A(1) und B(1) \+ geplottet werden - f\374r " }{TEXT 279 1 "t" }{TEXT -1 9 " =1 z.B.)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ka:='ka': kb:='kb':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "contourplot(\{ka*A('t')+kb*B('t')\},kb=0..20,ka= 0..5,axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 510 510 {PLOTDATA 2 "6,-%'CURVESG6hn7$7$$\"33+++++++;!#<$\"3gS\"oUWW%>')!#=7$$\"3C;n!GUF $R8F*$\"\"\"\"\"!7$7$$\"3Y;n!GUF$R8F*F17$$\"3%*4j3HxXu7F*$\"3aA%Gxc&Q \"3\"F*7$7$$\"3I:e\"F*F87$F>7$$\"3.`Gy(p T59\"F*$\"3y'GabdRZJ\"F*7$7$$\"3vz5@cGA>6F*$\"3!**************R\"F*FD7 $FJ7$$\"3T4)\\hj#=P6F*$\"3ZZD'4M/d^\"F*7$7$$\"3A&3J:IEnC\"F*$\"3#)**** *********z\"F*7$$\"3mr_9fqp\"H\"F*$\"3)>o8_tvq(=F*7$7$$\"3Ew:=**H1%Q\" F*$\"3c**************>F*Fen7$F[o7$$\"3b&zf:]9jT\"F*$\"3'305YP@f/#F*7$7 $$\"3s;T;.=wd:F*$\"3s*************>#F*Fao7$7$$\"3%p6kJ!=wd:F*Fjo7$$\"3 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$\"3/o$e=xtqc)F*F]j]l7$7$Fg]l$\"35RoO%)QNHeF^_uF_j]l7$7$Fg]l$\"3)=%oO% )QNHeF^_u7$$\"3_E`iWe#oO*F*Fhj]l7$7$Fd^l$\"3X;;U6DwMeF^_uFjj]l7$7$Fcfp $\"3%yh@9^iZ$eF^_u7$$\"3^8V&*\\4m;5F\\`lFc[^l7$7$Fj_l$\"3!Hc&HM\"\\%Re F^_uFe[^l7$7$Fj_l$\"3)\\c&HM\"\\%ReF^_u7$$\"3uBpQ![^VeF^_uF`\\^l7$7$Fjhp$\"3q5\"p@8#yp\\F*7$$\"3'31Fi)*=e 2\"F\\`lF`jr7$7$F_aw$\"3LL#pQ![^VeF^_u7$$\"37BXySfiw6F\\`lF`]^l7$7$Fgj m$\"3C%*QwV*eq%eF^_uFb]^l7$7$Faim$\"3!)4\"p@8#yp\\F*7$$\"3^&*)**ftvc8 \"F\\`l$\"3#GE++m53'\\F*7$7$Fgjm$\"30#y$HQ%4d#\\F*F]^^l7$7$Fgjm$\"3-(* QwV*eq%eF^_u7$$\"3oW\\Ak(e[ F*Fd`^l7$7$$\"39++++++g8F\\`l$\"3ob(*)y,9H&eF^_u7$$\"3CT%GRM)e;9F\\`lF aa^l7$7$F^`n$\"3[r@21^NbeF^_uFca^l7$7$F_a^l$\"37\"H$o'G>(e[F*7$$\"3b)* *Qm_*3D9F\\`l$\"3k1DS$=ws$[F*7$7$F^`n$\"3k)QI*yTnK[F*F^b^l7$Fga^l7$$\" 3u6ddz&yl\\\"F\\`lFha^l7$7$F[bn$\"3A#=N4jQv&eF^_uFhb^l7$Fdb^l7$$\"3tg5 ^))RV::F\\`l$\"3%4NA(G]T6[F*7$7$F[bn$\"3ILvNNZ@5[F*F`c^l7$F\\c^l7$$\"3 gfiZX)pld\"F\\`lF]c^l7$7$Fdcn$\"3Ycc;F\\`lF[e^l7$7$F]en$\"3=Kzo_#z7'eF^_uF]e^l7$Fbd^l7$$\" 3Y^:c7xa.;F\\`l$\"3#Q7'f=28\"z%F*7$7$F]en$\"39m&*f]U&ox%F*Fee^l7$Fae^l 7$$\"3f[#*)H)[bO6w%F*7$7$Fign$\"3Q_ %\\>,C53erleF^_uFih^l7$Fbh^l7$$\"3+x$)*f53erleF^_u7$$\"3C$fnn 8Pl(>F\\`lF\\j^l7$7$F[[o$\"3oNMu#*o&p'eF^_uF^j^l7$Fgi^l7$$\"3I#**>D=J[ %>F\\`l$\"3YA+qV?#zt%F*7$7$F[[o$\"3]k^K!Rn9t%F*Ffj^l-Fial6&F[blF2F3F3- %*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6$%#kbG%#kaG" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Wem zwei DGn nicht genug sind, d er kann den Zerfall leicht ein paar Generationen weiter verfolgen:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unassign('A','B','C','t'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dgln:=diff(A(t),t)=-ka* A(t)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " diff(B(t),t)= ka *A(t)-kb*B(t)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " diff(C( t),t)= kb*B(t)-kc*C(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%dglnG6%/-%%diffG6$-%\"AG6#%\"tGF-,$*&%#kaG\"\"\"F*F1!\"\"/-F(6$-%\" BGF,F-,&F/F1*&%#kbGF1F6F1F2/-F(6$-%\"CGF,F-,&F9F1*&%#kcGF1F>F1F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# diff(X(t),t)= \+ kc*C(t)-kx*X(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "\nsol:=dsolve(\{dgln\},\{A(t),B(t),C(t)\},method=lapl ace);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<%/-%\"AG6#%\"tG*&-F(6 #\"\"!\"\"\"-%$expG6#,$*&%#kaGF/F*F/!\"\"F//-%\"BGF),&*&-F16#,$*&%#kbG F/F*F/F6F/-F9F-F/F/**,&F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Sp \344testens hier lohnt sich der Einsatz eines CAS!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 5 "Aber:" }}{PARA 0 "" 0 "" {TEXT -1 208 "F\374r Plots und Berechnungen: Eingabe konkreter Zahlen \+ nur ganzzahlig oder in wissenschaftlicher Notation, wenn die Anzahl de r Tochterelemente 4 \374bersteigt (MapleV4-Bug bei der Verarbeitung v on Dezimalzahlen)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "Eigenwert" {TEXT 281 24 "Eigenwe rte und -vektoren" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 335 "Aber es l\344\337t sich mit Maple mehr machen, als lange Formeln zu produzieren. Wir k\366nnen bei dieser Gelegenheit die alge braische Behandlung von DG-systemen untersuchen (1. Ordnung, linear mi t konst. Koeff.), weil dies ein wichtiges Hilfsmittel f\374r die L\366 sung von DGn h\366herer Ordnung ist. Ein solches (homogenes) System ha t immer die Form " }{TEXT 282 3 "y' " }{TEXT -1 2 "= " }{TEXT 283 3 "M y" }{TEXT -1 8 ", wobei " }{TEXT 284 1 "M" }{TEXT -1 21 " eine Matrix ist und " }{TEXT 285 2 "y'" }{TEXT -1 5 " und " }{TEXT 286 1 "y" } {TEXT -1 16 " Vektoren. W\344re " }{TEXT 287 1 "M" }{TEXT -1 84 " ein \+ Skalar, so k\366nnten wir die L\366sung auch ohne Maple sofort hinschr eiben, n\344mlich " }{TEXT 288 1 "y" }{TEXT -1 4 " = " }{TEXT 289 4 " y(0)" }{TEXT -1 4 "exp(" }{TEXT 290 1 "M" }{TEXT -1 1 " " }{TEXT 291 1 "t" }{TEXT -1 53 "). Aber es ist ja denkbar, da\337 es bestimmte Vek toren " }{TEXT 292 2 "y0" }{TEXT -1 13 " und Skalare " }{TEXT 293 1 "k " }{TEXT -1 21 " gibt, f\374r die gilt: " }}{PARA 0 "" 0 "" {TEXT 294 1 "M" }{TEXT -1 1 " " }{TEXT 295 3 "y0 " }{TEXT -1 2 "= " }{TEXT 296 4 "k y0" }{TEXT -1 99 ". Versuchen wir es am \374bersichtlichen Beispi el von zwei Gleichungen (zwei radioaktiven Substanzen)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ka:='ka':kb:='kb':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "M:=matrix(2,2,[-ka,0,ka,-kb]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"MGK%'matrixG6#7$7$,$%#kaG!\"\"\"\"!7$F+,$%#kbGF,Q (pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Wenn obige Gleichu ng gelten soll, dann mu\337 die Determinante von" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Md:=matrix(2,2,[-ka-k,0,ka,-kb-k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MdGK%'matrixG6#7$7$,&%#kaG!\"\"%\"kGF,\" \"!7$F+,&%#kbGF,F-F,Q(pprint16\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "verschwinden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(Md) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%#kaG!\"\"%\"kGF&\"\"\",&%#kb GF&F'F&F(" }}}{EXCHG {PARA 0 "" 0 "Polynom, charakteristisches" {TEXT -1 50 "Die Nullstellen dieses charakteristischen Polynoms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "charpoly(M,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"kG\"\"\"%#kaGF&F&,&F%F&%#kbGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "sind die Eigenwerte von M:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$%#kaG!\"\",$%#kbGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Wir ben\366tigen noch die passenden Vektoren (" }{TEXT 297 2 "y 0" }{TEXT -1 5 " = ( " }{TEXT 298 2 "y1" }{TEXT -1 2 ", " }{TEXT 299 2 "y2" }{TEXT -1 29 ")) zu den beiden Eigenwerten " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(student): # fuer equate" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sys:=equate(evalm(M&*[y1,y2]),evalm (-ka*[y1,y2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<$/,$*&%#kaG \"\"\"%#y1GF*!\"\"F'/,&F(F**&%#kbGF*%#y2GF*F,,$*&F)F*F1F*F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(sys,\{y1,y2\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#y1G,$*(%#y2G\"\"\",&%#kaGF)%#kbG! \"\"F)F+F-F-/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "sys:=e quate(evalm(M&*[y1,y2]),evalm(-kb*[y1,y2]));solve(sys,\{y1,y2\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<$/,$*&%#kaG\"\"\"%#y1GF*!\"\", $*&%#kbGF*F+F*F,/,&F(F**&F/F*%#y2GF*F,,$F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#y2GF%/%#y1G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Aber es gen\374gt ein einziger Maple-Befehl:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,$%#kaG!\"\"\"\"\"<#K%'vectorG6#7$,$*&,&F%F'%#kbGF&F' F%F&F&F'Q(pprint26\"7%,$F0F&F'<#KF*6#7$\"\"!F'Q(pprint3F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Dabei steht an erster Stelle der Eigenwert, dann seine Vielfac hheit und in \{\} der Eigenvektor. Nun k\366nnen wir unser DG-System a lso einfacher schreiben: " }{TEXT 300 3 "y0'" }{TEXT -1 3 " = " } {TEXT 301 1 "k" }{TEXT -1 1 " " }{TEXT 302 2 "y0" }{TEXT -1 5 " mit " }{TEXT 303 1 "k" }{TEXT -1 3 " = " }{TEXT 304 3 "-ka" }{TEXT -1 6 " od er " }{TEXT 305 3 "-kb" }{TEXT -1 35 " und den zugeh\366rigen Eigenvek toren " }{TEXT 306 2 "y0" }{TEXT -1 76 ". Damit diese Eigenvektoren da s DG-System l\366sen, m\374ssen sie die Form haben: " }{TEXT 307 3 "y0 " }{TEXT -1 3 "= (" }{TEXT 308 2 "y1" }{TEXT -1 2 ", " }{TEXT 309 2 " y2" }{TEXT -1 6 ") exp(" }{TEXT 310 1 "k" }{TEXT -1 1 " " }{TEXT 311 1 "t" }{TEXT -1 138 "). Damit l\366st aber auch jede Linearkombination der beiden L\366sungen das System, insbes. erh\344lt man nach Multipl ikation obiger Vektoren mit " }}{PARA 0 "" 0 "" {TEXT 312 2 "ka" } {TEXT -1 2 "/(" }{TEXT 313 2 "kb" }{TEXT -1 1 "-" }{TEXT 314 2 "ka" } {TEXT -1 1 ")" }{TEXT 315 4 "A(0)" }{TEXT -1 15 " die schon mit " } {TEXT 316 6 "dsolve" }{TEXT -1 31 " erzeugte L\366sung A(t) und B(t). " }}{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 "Exponentialfunktion" {TEXT 317 33 "Die Rolle der \+ Exponentialfunktion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "Differenzengleichung" {TEXT -1 293 "Wir haben bisher die Propo rtionalit\344t der Exponentialfunktion zu ihrer eigenen Ableitung rein formal verwendet. Aber es gibt einen elementaren Zugang zu diesem Sac hverhalt. Zu diesem Zweck gehen wir einen Schritt \"zur\374ck\", n\344 mlich von der Differential- zur Differenzengleichung. Dann wird aus " }{TEXT 318 2 "y'" }{TEXT -1 2 " " }{TEXT 319 25 "Dy/Dt = (y(t+Dt)-y(t ))/Dt" }{TEXT -1 77 " und wenn wir das Zeitintervall 1 w\344hlen k\366 nnen wir die aufeinanderfolgenden " }{TEXT 320 1 "y" }{TEXT -1 18 "-We rte numerieren:" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "dg:=y(n+1)-y(n)=k*y(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dgG/,&-%\"yG6#,&%\"nG\"\"\"F,F,F,-F(6#F+!\"\"*&%\"kG F,F-F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Diese rekursive Beziehu ng hat bekanntlich die L\366sung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sol:=rsolve(dg,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$solG*&-%\"yG6#\"\"!\"\"\"),&F*F*%\"kGF*%\"nGF*" }}}{EXCHG {PARA 0 "" 0 "Folge, geometrische" {TEXT -1 33 "also eine geometrische Folge m it " }{TEXT 321 1 "q" }{TEXT -1 7 " = 1 + " }{TEXT 322 1 "k" }{TEXT -1 33 ". Wir k\366nnen zu beliebiger Basis " }{TEXT 323 1 "q" }{TEXT -1 166 " Punktplots dieser Folgen erstellen, auch zu negativer Basis. \+ Dann wird Wachstum und Zerfall gemischt und wir bekommen Schwingungen \+ -- ged\344mpfte, anwachsende und f\374r " }{TEXT 324 2 "q " }{TEXT -1 170 "= -1 etwas, was wie eine Sinusschwingung aussieht. Und das obwohl wir \"nur\" eine DG / Differenzengl. 1. Ordnung haben! Setzen Sie als o f\374r den n\344chsten Plot verschiedene " }{TEXT 325 1 "q" }{TEXT -1 20 " ein, auch negative:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "q:=4/5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([seq([n, q^n],n=0..10)],style=point,color=red);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 510 510 {PLOTDATA 2 "6'-%'CURVESG6#7-7$$\"\"!F)$\"\"\"F)7$F*$\"3U+ ++++++!)!#=7$$\"\"#F)$\"39+++++++kF/7$$\"\"$F)$\"35++++++?^F/7$$\"\"%F )$\"3?++++++'4%F/7$$\"\"&F)$\"3G+++++!oF$F/7$$\"\"'F)$\"3))*********R9 i#F/7$$\"\"(F)$\"3!*********>:(4#F/7$$\"\")F)$\"3/++++;sx;F/7$$\"\"*F) $\"33+++!Gx@M\"F/7$$\"#5F)$\"3/+++C=ut5F/-%&STYLEG6#%&POINTG-%'COLOURG 6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$Q!6\"Feo-%%VIEWG6$%(DEFAULTG Fjo" 1 5 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Aber warum ist der Plot-Befe hl so umst\344ndlich formuliert? Weil Maple f\374r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot((-1)^n,n=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 510 510 {PLOTDATA 2 "6%-%'CURVESG6$7%7$$\"\"!F)$\"\"\"F)7$%*undefi nedGF-7$$\"#5F)F*-%'COLOURG6&%$RGBG$F0!\"\"F(F(-%+AXESLABELSG6$Q\"n6\" Q!F;-%%VIEWG6$;F(F/%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "nur ein leere s Koordinatensystem zeigt. Woran liegt das? " }}{PARA 0 "" 0 "" {TEXT -1 26 "Maple versucht z.B. auch " }{XPPEDIT 18 0 "(-1)^(1/2" "6#),$\" \"\"!\"\"*&F%F%\"\"#F&" }{TEXT -1 60 " zu berechnen und das ist nicht mehr reell. 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Negative Basis bedeutet also \+ im allgemeinen Fall ein komplexes Argument der Exponentialfunktion (" }{TEXT 327 1 "q" }{TEXT -1 6 " < 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "'q^t'='(-abs(q))^t','exp(Pi*I*t)*exp(ln(abs(q))*t)'=' exp((Pi*I+ln(abs(q)))*t)';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/)%\"qG% \"tG),$-%$absG6#F%!\"\"F&/*&-%$expG6#*(%#PiG\"\"\"F&F4^#F4F4F4-F06#*&F &F4-%#lnG6#F)F4F4-F06#*&,&*&F3F4F5F4F4F9F4F4F&F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Wenn \+ wir nun von der Differenzengleichung und der geometrischen Folge zur D G " }{TEXT 328 2 "y'" }{TEXT -1 3 " = " }{TEXT 329 1 "k" }{TEXT -1 1 " " }{TEXT 330 1 "y" }{TEXT -1 49 " zur\374ckkehren, bedeutet das komp lexe Eigenwerte " }{TEXT 331 1 "k" }{TEXT -1 493 ". Ist das nur eine S pielerei? Nein, diese Zusammenh\344nge reichen bis in die Quantenphysi k: w\374rde man bei reellen Exponenten (Eigenwerten) bleiben, k\344me \+ man nie vom Zerfall / Wachstum zur Schwingung, nicht von der aperiodis chen und inkoh\344renten Diffusion zur periodischen und koh\344renten \+ Welle und auch nicht zur Schr\366dingergleichung, die ja in der Zeit e ine DG 1.Ordng. ist. Die \"imagin\344re Einheit\" scheint eine tiefer \+ liegende Wirklichkeit zu beschreiben ... [vgl. Reihenentwicklung von e xp()] " }}{PARA 0 "" 0 "" {TEXT -1 96 "Bei dieser Gelegenheit k\366nne n wir noch die Exponentialfunktion mit komplexem Argument darstellen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "q:=-4/5: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "rq:=evalc(Re(q^t)):iq:=evalc(Im(q^t )):aq:=evalc(abs(q^t)): #beschleunigt den Plot" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "plot(\{rq,iq,aq\},t=0..5);" }}{PARA 13 "" 1 " " {GLPLOT2D 510 510 510 {PLOTDATA 2 "6'-%'CURVESG6$7fq7$$\"\"!F)$\"\" \"F)7$$\"3GLLL3x&)*3\"!#=$\"3Ezk>KwB$>*F/7$$\"3))**\\i!R(*Rc\"F/$\"3#e 1^W;#f9&)F/7$$\"3umm\"H2P\"Q?F/$\"3!*[K$)>uiiwF/7$$\"3!***\\PMnNrDF/$ \"3I+iD7FUDlF/7$$\"3MLL$eRwX5$F/$\"3aE`p^D_L_F/7$$\"3rLLL$eI8k$F/$\"3e Zqwkt'o\"QF/7$$\"33ML$3x%3yTF/$\"3h6^]\\u@EBF/7$$\"3h+]PfyG7ZF/$\"3#e4 F0m$\\D\")!#>7$$\"3emm\"z%4\\Y_F/$!3MM#4%*=R8)oFU7$$\"32++v$flE.'Q'eJ?F/7$$\"3`LLeR-/PiF/$!3)H;m0^yoH$F/7$$\"31nm\"zWo)\\nF/$!3wS* \\W)45%\\%F/7$$\"3]***\\il'pisF/$!3gc\"fP@y&[bF/7$$\"3Qnm\"HKkIz(F/$!3 _#yLW7xNY'F/7$$\"3>MLe*)>VB$)F/$!3]!)*HD$fNzrF/7$$\"3wmmTg()4_))F/$!3) >MuFLG'zwF/7$$\"3Y++DJbw!Q*F/$!3y#\\m\"okIezF/7$$\"3E$3_+A:n^*F/$!3mJo U%37P*zF/7$$\"3=nT&)3\\m_'*F/$!3dco(y)\\H9!)F/7$$\"33^il(f9')y*F/$!3c# *y7Cs5?!)F/7$$\"3+N$ekGkX#**F/$!3)Q%*Ru^F7,)F/7$$\"3nTg_(R^g+\"!#<$!3] skZl4w()zF/7$$\"31]iSmjk>5Fdr$!3A'33xoQ)\\zF/7$$\"3XekGN8CL5Fdr$!3=&*z QGuh(*yF/7$$\"3%ommTIOo/\"Fdr$!3t*f*4:\"z7$yF/7$$\"3E+]7GTt%4\"Fdr$!3# e8<>EB$)[(F/7$$\"3YLL3_>jU6Fdr$!3)HmtA\"Fdr$!3CU _t'opFC'F/7$$\"37++]i^Z]7Fdr$!3t&p`B#ROT`F/7$$\"35+++v\"=YI\"Fdr$!3kkT -aJ30VF/7$$\"33++](=h(e8Fdr$!3%)\\O]1s:qJF/7$$\"3&*****\\7!Q4T\"Fdr$!3 \\D`1uVq:?F/7$$\"3/++]P[6j9Fdr$!3L7>#ex&RT$)FU7$$\"3%o;HKR'\\5:Fdr$\"3 a-c)RN!e`BFU7$$\"3UL$e*[z(yb\"Fdr$\"3%)Q7'*R6Ix7F/7$$\"34+Dc,#>Uh\"Fdr $\"3-4Bk\"o(f\\CF/7$$\"3wmm;a/cq;Fdr$\"3kBRsEY\"o^$F/7$$\"3\"pm;a)))G= Fdr$\"3Q-&o:45@B'F/7$$\"3KLe9;0?E>Fdr$\"3C8T8d<=KjF/7$$\"3pTg-gl[Q> Fdr$\"3[Z!oB2twO'F/7$$\"31]i!Rgs2&>Fdr$\"3ycQ@lKY$R'F/7$$\"3WekyZ'eI'> Fdr$\"34878xoc4kF/7$$\"3gmmm\"pW`(>Fdr$\"3u2#3B$G,;kF/7$$\"3_ek.HW#))) >Fdr$\"3DNbqTg-7kF/7$$\"3?]iSmTI-?Fdr$\"3+mQ-JEa'R'F/7$$\"3*=/wP!Ry:?F dr$\"3)[uWNPf'pjF/7$$\"3dLe9TOEH?Fdr$\"3sm(4V%H\\JjF/7$$\"3'pT&)e6Bi0# Fdr$\"3H%ziBxu=A'F/7$$\"3K+]i!f#=$3#Fdr$\"3[Xe%eA1!pgF/7$$\"3/++D\"=EX 8#Fdr$\"3G?&pU#HGkcF/7$$\"3?+](=xpe=#Fdr$\"3\\EpuTYhA^F/7$$\"37nm\"H28 IH#Fdr$\"3UQ)=zdJ$HOF/7$$\"3$p;a8d3AM#Fdr$\"3h0**z^uU?GF/7$$\"3um;zpSS \"R#Fdr$\"3!*y$p!y:Di>F/7$$\"3-+v$41oWW#Fdr$\"3\"yjA`#3*f+\"F/7$$\"3GL L3_?`(\\#Fdr$\"3JVsVK3kSW!#?7$$\"3AL3_D1l_DFdr$!3\"ReJ8!eB:$*FU7$$\"3f L$e*)>pxg#Fdr$!3KF'y:szg&=F/7$$\"3%omm\"z+vbEFdr$!3a$>!Q'>*f)f#F/7$$\" 33+]Pf4t.FFdr$!3GEP]#e)ymKF/7$$\"3uLLe*Gst!GFdr$!3'R!GDa2p&R%F/7$$\"3) om\"H2\"34'GFdr$!3MEVSlF:&y%F/7$$\"30+++DRW9HFdr$!31C=kE$48.&F/7$$\"3S +Dc,6jSHFdr$!3'H<]([()H)4&F/7$$\"3K+]7y#=o'HFdr$!315.-F00I^F/7$$\"3G]i Sm=\"*zHFdr$!37^7LdRwK^F/7$$\"3C+voaa+$*HFdr$!3(*[#=qZfn7&F/7$$\"3>](o H/*41IFdr$!3-rekg!)47^F/7$$\"3:++DJE>>IFdr$!3gKS]/]&))3&F/7$$\"3A+v$4^ n)pIFdr$!3QbWK>V&)>\\F/7$$\"3F+]i!RU07$Fdr$!32NY6:8)4j%F/7$$\"3+++v=S2 LKFdr$!3GF#eoz=Zh$F/7$$\"3Jmmm\"p)=MLFdr$!3Ugo&[Xs\\O#F/7$$\"3B++](=]@ W$Fdr$!3Ujhdpqa%Q)FU7$$\"35L$e*[$z*RNFdr$\"3E2p\\Spp&o&FU7$$\"3e++]iC$ pk$Fdr$\"3Y]t6.\"GQ(>F/7$$\"3[m;H2qcZPFdr$\"3^X%*[luiSIF/7$$\"3UL$3_0j ,!QFdr$\"3maJzFR8mMF/7$$\"3O+]7.\"fF&QFdr$\"3CMJ$*Q)yzy$F/7$$\"3)oT&QG -ZyQFdr$\"32ULNTTc0RF/7$$\"3TLek`8=/RFdr$\"3cl++.\"ej*RF/7$$\"3$*\\i!* yC*)HRFdr$\"3'o3x[uu+1%F/7$$\"3Ymm;/OgbRFdr$\"3e9(fXz2m4%F/7$$\"3Y\\7` p2_#)RFdr$\"3@rizN!4e5%F/7$$\"3*G$e*[$zV4SFdr$\"3ISV?,je&3%F/7$$\"3L;/ E+^NOSFdr$\"3LCKkB`VOSF/7$$\"3w**\\ilAFjSFdr$\"3#[1bWX_!fRF/7$$\"3ym\" zW7@^6%Fdr$\"3qN\"z;MnQt$F/7$$\"3yLLL$)*pp;%Fdr$\"3%*HB0.hi:MF/7$$\"3) RL$3xe,tUFdr$\"31.J#*y[8@DF/7$$\"3Cn;HdO=yVFdr$\"3i#Hqv-kdS\"F/7$$\"3a +++D>#[Z%Fdr$\"3Fysy1'[6\"HFU7$$\"3SnmT&G!e&e%Fdr$!3y#RDYW5va*FU7$$\"3 #RLLL)Qk%o%Fdr$!3Mf]17x!p#>F/7$$\"37+]iSjE!z%Fdr$!3#ooG9w&*\\r#F/7$$\" 3L+++DM\"3%[Fdr$!3;7w()zV[zHF/7$$\"3a+]P40O\"*[Fdr$!3Cv\"y3yjN;$F/7$$ \"3>]7.#Q?&=\\Fdr$!3>bd\"**f%>GKF/7$$\"3s+voa-oX\\Fdr$!3'47@c_#eoKF/7$ $\"3O]PMF,%G(\\Fdr$!3!)os-ugs%G$F/7$$\"\"&F)$!3#3+++++oF$F/-%'COLOURG6 &%$RGBG$\"#5!\"\"F(F(-F$6$7`r7$F(F(7$$\"3ALL$3FWYs#FU$\"3)yE&z1GX(\\)F U7$$\"3WmmmT&)G\\aFU$\"35CB@gZ+$o\"F/7$$\"3m****\\7G$R<)FU$\"31>))>(e# )Q\\#F/7$F-$\"3_G_t%)yrwKF/7$F3$\"3a)QO=T&HcXF/7$F8$\"34RD<$fJ(3dF/7$F =$\"3+s\"f2r[Z#oF/7$FB$\"3WU2#[$4wCxF/7$FG$\"3!)HB4C]S#R)F/7$FL$\"3Qyd .C\">y!))F/7$$\"3H](oHaN;J%F/$\"3oF4&)>A;r))F/7$$\"31nT5:j=XWF/$\"38fA A&)H[=*)F/7$$\"3#QeRs3P(yXF/$\"3)3<\"etJz\\*)F/7$FQ$\"3Q)G$41F8l*)F/7$ $\"3Q[*)F/ 7$$\"3#)\\7yv,%H6&F/$\"3i#G-%oP9;*)F/7$FW$\"3!)[())=#Rbo))F/7$Ffn$\"39 <#))>ra'f&)F/7$F[o$\"3/p46Fs$>0)F/7$F`o$\"3B]VV]mQMtF/7$Feo$\"3fwx&f\\ @VW'F/7$Fjo$\"3m13>44!4P&F/7$F_p$\"3#[`Im(G$[<%F/7$Fdp$\"3_JbQ^Y5'*GF/ 7$Fip$\"3gNW!*zd-o:F/7$F]r$\"3ceXOrD#*)*=FU7$Fbs$!3u-![$pvog6F/7$Fgs$! 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