\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fmodern\fprq1 Courier New;}{\f5\froman\fcharset1 Times New Roman;}{\f6\fswiss\fprq2 Verdana;}{\f7\froman\fprq2 Times New Roman;}{\f8\fswiss\fprq2 Helvetica;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red0\green128\blue0;\red255\green0\blue0;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f4\fs20\cf0 Inhalt....: Das Euler-Cauchy Verfahren zur L\'f6sung von Differenzialgleichung \par Kategorie.: Arbeitsblatt \par Mathematik: Analysis, Numerik \par MuPAD.....: 3.0.0 \par Datum.....: 2002-03-06 \par Autoren...: Agnes Havasi \par Autoren...: Kai Gehrs \par Funktionen: diff, plotfunc2d, solve, simplify, subs, plot, plot::Line2d, \par Funktionen: plot::Point2d, plot::VectorField2d, plot::Ode2d, PointSize \par Funktionen: numeric::ode2vectorfield, ode \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs36\cf0\b \par \plain\f3\fs40\cf0\b Das Euler-Cauchy Verfahren zur L\'f6sung von \par Differenzialgleichungen \plain\f3\fs24\cf2 \par \par \plain\f3\fs28\cf0 \par \plain\f3\fs24\cf2 In diesem Notebook stellen wir das Euler-Cauchy-Polygonzug Verfahren zur approximativen \par grafischen L\'f6sung von Differenzialgleichungen vor. Der mathematische Inhalt geht stellenweise \par recht entscheidend \'fcber das hinaus, was standardm\'e4ssig in Schulen behandelt werden kann. \par Die Motivation bedient sich Beispielen der Physik, die ebenfalls recht komplex sind. \par Es ist jedoch \plain\f3\fs24\cf2\i nicht\plain\f3\fs24\cf2 \plain\f3\fs24\cf2\i unbedingt notwendig\plain\f3\fs24\cf2 , ein physikalisch bzw. mathematisch \par fundiertes Wissen auf dem Gebiet der gew\'fchnlichen Differenzialgleichungen zu besitzen, \par um die wesentlichen mathematischen Ideen, die hier vorgestellt werden, nachvollziehen \par zu k\'f6nnen. Vielmehr sollte man das Notebook als Appetithappen oder als eventuelle \par Anregung f\'fcr Facharbeiten bzw. mathematisch technisch interessierte Sch\'fclerinnen und \par Sch\'fcler bzw. Studentinnen und Studenten verstehen. \par \plain\f3\fs28\cf0 \par \plain\f3\fs28\cf0\b Motivation: \par \plain\f3\fs28\cf0 \par Differenzialgleichungen treten in vielen Bereichen der Natur und Technik auf. \par So wird zum Beispiel die Bewegung einer an einer elastischen Feder \par aufgeh\'e4ngten Kugel der Masse m beschrieben durch die Gleichung \par \plain\f3\fs28 {\pict\wmetafile8\picw5086\pich1077\picscalex99\picscaley99\picwgoal2909\pichgoal613 0100090000032104000009001C0000000000050000000B0200000000050000000C023504DE1303 0000001E00050000000C023A040C14050000000B0200000000030000001E00050000000C024104 1314050000000B0200000000050000000B0200000000030000001E00050000000C024704411405 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C024E044814050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C0255047814050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000030000001E 00050000000C025C047F14050000000B0200000000050000000B0200000000050000000B020000 0000050000000B0200000000050000000B0200000000050000000B0200000000030000001E0005 0000000C026304AF14050000000B0200000000050000000B0200000000050000000B0200000000 050000000B0200000000050000000B0200000000050000000B0200000000050000000B02000000 00030000001E00050000000C026904B614050000000B0200000000050000000B02000000000500 00000B0200000000050000000B0200000000050000000B0200000000050000000B020000000005 0000000B0200000000050000000B0200000000030000001E00030000001E00050000000C028002 BE0B050000000B0200000000050000000B0200000000050000000B0200000000050000000B0200 000000050000000B0200000000050000000B0200000000050000000B0200000000050000000B02 00000000050000000B020000000008000000FA0200000000000000000000040000002D0100001C 000000FB0238FF00000000000090010000000107000000417269616C000000D40F0A7138E91200 D89FF177E19FF1772020F377CA0F6695040000002D010100050000000201010000000500000001 02FFFFFF00050000002E01180000000500000009020000000004000000080100001C000000FB02 E8FE0000000000009001000000010700000054696D6573204E657720526F6D616E00D89FF177E1 9FF1772020F377CA0F6695040000002D0102000B00000026060F000C004D617468547970650000 CD001C000000FB02E8FE0000000000009001010000010700000054696D6573204E657720526F6D 616E00D89FF177E19FF1772020F377CA0F6695040000002D0103001C000000FB023AFF00000000 00009001000000010700000054696D6573204E657720526F6D616E00D89FF177E19FF1772020F3 77CA0F6695040000002D010400040000002D0103001C000000FB02E8FE00000000000090010000 0002070000005346204D617468204578740038E91200D89FF177E19FF1772020F377CA0F669504 0000002D010500040000002D010200040000002D010500040000002D010200040000002D010300 040000002D010500040000002D010200040000002D010500040000002D0102001C000000FB02E8 FE0000000000009001000000020700000053796D626F6C0000D50F0A5D38E91200D89FF177E19F F1772020F377CA0F6695040000002D010600040000002D010300040000002D010200040000002D 010300040000002D010200040000002D010300040000002D010200040000002D01060004000000 2D010200040000002D010600040000002D010200040000002D010400040000002D010200040000 002D0104000500000009020000FF00040000002D010300070000002105010079007F017B000400 00002D01040007000000210501002700420114010700000021050100270042015301040000002D 010500040000002D010200040000002D010500040000002D010200040000002D01050007000000 2105010028007F017501040000002D010600040000002D010300070000002105010074007F01E2 01040000002D010500040000002D010200040000002D010500040000002D010200040000002D01 0500070000002105010029007F013502040000002D01060007000000210501003D007F01F60204 0000002D01030007000000210501006B007F01E403040000002D0106000700000021050100D700 7F01A504040000002D010300070000002105010079007F013A05040000002D010500040000002D 010200040000002D010500040000002D010200040000002D010500070000002105010028007F01 B605040000002D010600040000002D010300070000002105010074007F012306040000002D0105 00040000002D010200040000002D010500040000002D010200040000002D010500070000002105 010029007F017606040000002D01060007000000210501002C007F01E306040000002D01030007 000000210501006B007F018B07040000002D01060007000000210501003D007F01680807000000 210501002D007F018009040000002D010200040000002D01030007000000210501006300FF0098 0A07000000210501006D001902710A040000002D0105000700000021050100C5002101520A0700 000021050100C5002101B20A08000000FA0200000000000000000000040000002D0107001C0000 00FB021000070000000000BC02000000000102022253797374656D0000CB0F0AB138E91200D89F F177E19FF1772020F377CA0F6695040000002D010800040000002701FFFF04000000F001000004 000000F001010004000000F001020004000000F001030004000000F001040004000000F0010500 04000000F0010600040000002701FFFF040000002701FFFF040000002701FFFF040000002701FF FF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701 FFFF030000000000 }\plain\f3\fs28 \par \plain\f3\fs28\cf0 wobei y(t) den Ort der Kugel zur Zeit t beschreibt, y''(t) ihre Beschleunigung \par und k eine Konstante ist. 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}\plain\f3\fs28 \par Diese Gleichung ist eine sogenannte Differenzialgleichung. In Differenzial- \par gleichungen tauchen Funktionen und ihre Ableitungen als Unbekannte auf. \par \par Alle Funktionen, die die obige Gleichung erf\'fcllen, m\'fcssen gleich ihrer \par ersten Ableitung sein. Wir kennen alle die e-Funktion aus dem Schul- \par unterricht. Leitet man diese Funktion ab, so erh\'e4lt man wieder die gleiche \par Funktion: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}y:= exp(t); y1:= diff(y, t) \par \pard\ri4\plain\f4\fs22\cf1 \par \pard\plain\f3\fs28 Diesen Sachverhalt kann man sich leicht anhand der folgenden beiden \par Witze merken:\plain\f6\fs28 \plain\f6\fs28\i \par \par \plain\f3\fs28\i Witze in der Mathematik \par \par Zwei Funktionen beschimpfen sich. \par Nach einem l\'e4ngeren Austausch von "Komplimenten", \par schlie\'dflich die eine: "Ich differenziere und integriere dich, \par bis du nicht mehr wei\'dft, wer du eigentlich bist!" \par Darauf antwortet die andere: "\'c4tsch, ich bin e hoch x !" \par \pard\ri4\plain\f3\fs28\cf1\i \par \pard\plain\f3\fs28\i Im Raum der stetigen Funktionen findet ein Tanzball statt. Auf der \par Tanzfl\'e4che tanzen Cosinus und Sinus auf und ab und die Polynome \par bilden einen Ring. Nur die Exponentialfunktion steht den ganzen Abend \par alleine herum. Aus Mitleid geht die Identit\'e4t irgendwann zu ihr hin und \par sagt: "Mensch, integrier dich doch einfach mal!" "Schon versucht!", \par antwortet die Exponentialfunktion, "Das hat aber auch nichts ge\'e4ndert!" \par \pard\ri4\plain\f4\fs22\cf1\i \par \plain\f3\fs28 Wir kennzeichnen im folgenden mit y1 immer die erste Ableitung der \par Funktion y. \par \par Ist die Funktion exp(x) die einzige L\'f6sung der obigen Differenzialgleichung? \par \par Die Antwort ist nein, denn f\'fcr eine beliebige reelle Zahl c gilt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}c * y = diff(c * y, t) \par \pard\ri4\plain\f3\fs28 \par d.h. alle Funktionen der Form y(x) = c * exp(x) erf\'fcllen die obige Differenzial- \par gleichung. \par \par Wie kann man die Bedingungen an eine L\'f6sung nun so weit versch\'e4rfen, \par dass die L\'f6sung eindeutig wird? \par \par Zun\'e4chst schauen wir uns einige der Kurven der Form c * exp(x) an: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}delete c: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 y_c:= c * exp(t) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}plotfunc2d(y_c $ c = -5..5, t = -1..1) \par \pard\ri4\plain\f3\fs28 \par Beobachtung: Jede Kurve hat mit der y-Achse einen anderen Schnittpunkt. \par Nun liegt es nahe, dass wir die L\'f6sungskurve durch eine Bedingung der \par Form y(0) = d f\'fcr eine reelle Zahl d "eindeutig machen", denn zu vorgegebenen \par d k\'f6nnen wir aus y(x) = c * exp(x) den entsprechenden Wert f\'fcr c berechnen. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}simplify(solve(d = c * subs(y, t = 0), c)) \par \pard\ri4\plain\f3\fs28 \par d.h. wir m\'fcssen c = d w\'e4hlen. \par \par BEISPIEL: d = 3 \par \plain\f4\fs28\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}simplify(solve(3 = c * subs(y, t = 0), c)) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}plotfunc2d(subs(y_c, c = 3), t = -1..1) \par \pard\ri4\plain\f3\fs28 \par Jetzt stellt sich die Frage: Wie l\'f6st man kompliziertere Gleichungen, wo man \par die L\'f6sung nicht wie oben "erraten" kann? \par \par Ein m\'f6gliches Verfahren ist das sogenannte Euler-Cauchy-Polygonzugverfahren. \par \par \par \plain\f3\fs28\b HISTORISCHES: \par \plain\f3\fs28 \par \pard\plain\f3\fs28\b Euler \plain\f3\fs28 (1707 - 1783)\plain\f3\fs15 \plain\f3\fs28 war einer der gr\'f6\'dften Mathematiker aller Zeiten. \par Er entwickelte die Grundlagen der modernen Zahlentheorie und Algebra, \par der Topologie, der Wahrscheinlichkeitsrechnung und Kombinatorik, der \par Integralrechnung, der Theorie der Diffenrentialgleichungen und der \par Differenzialgeometrie, der Variationsrechnung, entdeckte den Zusammenhang \par zwischen trigonometrischen Funktionen und Exponentialfunktionen. Leonard \par Euler entwickelte die Hydrodynamik und Str\'f6mungslehre, schuf die Grundlagen \par f\'fcr die Theorie des Kreisels. Er war ein genialer Naturwissenschaftler, ein \par hervorragender Lehrer und Mentor. \par \plain\f7\fs32 \par \plain\f3\fs28\b Cauchy\plain\f3\fs28 , geboren 21. 08. 1789 Paris, gestorben 22. 05. 1857 Sceaux (bei Paris). \par Cauchy, dessen Vater hohe administrative \'c4mter bekleidete, erhielt eine \par ausgezeichnete Privatausbildung. Er wollte Ingenieur werden und war auch \par einige Jahre nach Beendigung s\plain\f3\fs28\cf0 eines St\plain\f3\fs28 udiums als solcher t\'e4tig. Im \par Selbststudium eignete er sich Werke von LAGRANGE und P.S. LAPLACE an. \par Ab 1813 war Cauchy wieder in Paris. Seine mathematische Karriere begann aber \par schon 1811, als es ihm gelang ein Problem von LAGRANGE zu l\'f6sen. F\'fcr die \par L\'f6sung eines Problems der Hydromechanik bekam Cauchy 1816 den Preis der \par Pariser Akademie. \par Mit seinen Arbeiten zur Elastizit\'e4tstheorie geh\'f6rte er sp\'e4testens ab 1822 zu \par den herausragendsten Mathematikern seiner Zeit. In den Jahren 1815/16 war \par Cauchy Professor an der Ecole Polytechnique und wurde 1816 Mitglied der \par Franz\'f6sischen Akademie. Viele mathematische Begriffe und S\'e4tze sind mit \par seinem Namen verbunde, zum Beispiel der Satz von Cauchy, Cauchy-Folge, \par Cauchy-Riemannsche Differenzialgleichungen, Cauchysche Integralformel, \par Cauchyscher Integralsatz, Cauchysches Konvergenzkriterium. Cauchy geh\'f6rt \par zu den produktivsten Mathematikern aller Zeiten.\plain\f7\fs28 \par \pard\ri4\plain\f3\fs28 \par \plain\f3\fs28\b Zur\'fcck zur Mathematik an sich:\plain\f3\fs28 \par \par Die Idee des Verfahrens ist am leichtesten zu verstehen, wenn man sich \par ein Beispiel anschaut. 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00002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f3\fs28\cf1 \par \plain\f4\fs22\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}delete y, y1: \par {\pntext\f1\'b7\tab}y1:= t * y - y^2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}subs(y1, t = 0, y = 2) \par \pard\ri4\plain\f3\fs28 \par Wenn die Steigung bekannt ist, so k\'f6nnen wir die Gerade durch (0|2) mit \par Steigung -4 berechnen. \par \plain\f4\fs22\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}delete b: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 g:= -4 * x + b \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}solve( 2 = subs(g, x = 0),b) \par \pard\li600\ri1\fi-300\plain\f4\fs22\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}b:= 2 \par \pard\ri4\plain\f3\fs28 \par Die Gerade ist also \par \plain\f5\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}g \par \pard\ri4\plain\f3\fs28 \par Die Idee ist nun: Berechne den Funktionswert von g an der Stelle 1/4 (d.h. wir gehen \par einen viertel Schritt nach rechts): \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}subs(g, x = 1/4) \par \pard\ri4\plain\f3\fs28 \par Das bedeutet: y(1/4)= 1. \par \plain\f4\fs22\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}l1 := plot::Line2d([0, 2], [1/4, 1]): \par {\pntext\f1\'b7\tab}p1 := plot::Point2d([0, 2], PointSize = 3*unit::mm): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 p2 := plot::Point2d([1/4, 1], Color = RGB::Blue, \par PointSize = 3*unit::mm):\plain\f4\fs22\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}plot(l1,p1,p2) \par \pard\ri4\plain\f3\fs28 \par Setzen wir das Verfahren fort. Wir gehen von dem neuen Punkt (1/4|1) \par aus wieder um 1/4 nach rechts und berechnen die entsprechende Steigung \par und den Funktionswert wie oben: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}subs (y1, t = 1/4, y = 1) \par \pard\ri4\plain\f3\fs28 \par Die Steigung ist jetzt -3/4, die Gerade l\'e4uft durch (1/4|1). Wir bestimmen \par jetzt eine weitere Gerade h mit Steidung -3/4, die durch den Punkt (1/4|1) \par verl\'e4uft. \par \plain\f4\fs22\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}h:= -3/4 * x + c \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}solve(1 = subs(h, x = 1/4), c) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}c:= 19/16 \par \pard\ri4\plain\f3\fs28 \par Die Gerade sieht also folgenderma\'dfen aus: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}h \par \pard\ri4\plain\f3\fs28 \par Berechnen wir jetzt den Funktionswert von h an der Stelle 1/2, d.h. wir gehen \par noch einen viertel Schritt nach rechts: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}subs(h, x = 1/2) \par \pard\ri4\plain\f3\fs28 \par Also gilt n\'e4herungsweise: y(1/2) = 13/16 \par \par Zeichnen wir sie mit den bisherigen Ergebnissen zusammen in ein \par Koordinatensystem: \par \plain\f4\fs22\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p3:= plot::Point2d([1/2, 13/16], Color = RGB::Green, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 PointSize = 3*unit::mm): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}l2:= plot::Line2d([1/4, 1], [1/2, 13/16]): \par {\pntext\f1\'b7\tab}plot(p1,p2,p3,l1,l2) \par \pard\ri4\plain\f3\fs28 \par Wir setzen das Verfahren fort: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}subs (y1, t = 1/2, y = 13/16) \par \pard\ri4\plain\f3\fs28 \par Die Steigung der Gerade ist -65/256, die Gerade l\'e4uft durch den Punkt \par (1/2 | 13/16). Diese Gerade bezeichnen wir mit k: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}delete e: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 k:= -65/256 * x + e \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}solve( 13/16 = subs(k, x = 1/2) ,e) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}e:= 481/512 \par \pard\ri4\plain\f3\fs28 \par Die Geradengleichung is daher gegeben durch: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}k \par \pard\ri4\plain\f3\fs28 \par Berechnen wir jetzt den Funktionswert von k an der Stelle 3/4, d.h. wir \par gehen einen weiteren viertel Schritt nach rechts: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}subs(k, x = 3/4) \par \pard\ri4\plain\f3\fs28 \par d.h. y(3/4) = 767/1024. \par \par Zeichnen wir auch diese zusammen mit den vorherigen Ergebnissen: \par \plain\f4\fs28\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p4:= plot::Point2d([3/4, 767/1024], Color = RGB::Pink, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 PointSize = 3*unit::mm): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}l3:= plot::Line2d([1/2, 13/16], [3/4, 767/1024]): \par {\pntext\f1\'b7\tab}plot(p1,p2,p3,p4,l1,l2,l3) \par \pard\ri4\plain\f3\fs28 \par Wenn man dieses Verfahren immer weiter fortsetzt, so erh\'e4lt man einen \par Polygonzug, der bei hinreichend kleiner Schrittweite, eine Kurve ann\'e4hert. \par Dieser Polygonzug n\'e4hert dann die L\'f6sung der Differenzialgleichung \par zu der festen Anfangsbedingung y(0) = 2 an. \plain\f4\fs22\cf3 \par \plain\f3\fs28 \par Damit wir das Verfahren nicht in allen Schritten per Hand durchf\'fchren \par m\'fcssen, bietet MuPAD eine M\'f6glichkeit, zun\'e4chst die Steigung von \par Differenzialgleichungen in Form von Richtungsfeldern in der Ebene zu \par visualisieren. \par \par Sehen wir uns eine geometrische Veranschaulichung des Richtungsfelds \par der Differenzialgleichung y'(t) = t * y(t) - y(t) ^ 2 mit MuPAD an: \par \plain\f4\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}f := (t, y) -> t*y - y^2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}V1 := plot::VectorField2d([1, f(t, y)], t = 0..3, y = 0..3, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 Mesh = [10, 10], \par Color = RGB::Black): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}plot(V1) \par \pard\ri4\plain\f3\fs28\cf0 \par Wie interpretiert man das Bild? Man w\'e4hlt auf der y-Achse einen Anfangswert, \par z.B. y(0) = 2. Nun folgt man von diesem Punkt aus den Pfeilen in der Ebene. \par Dies liefert eine N\'e4herung der L\'f6sung der Differenzialgleichung zum gegebenen \par Anfangswert. Im obigen Beispiel, wo wir mit den Geraden gerechnet haben, \par haben wir die Pfeile mathematisch \'fcber die Steigung der berechneten Geraden \par angen\'e4hert. \par \par Wir zeichnen unsere N\'e4herungsl\'f6sung mit dem Richtungsfeld in ein \par Koordinatensystem: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}plot(p1,p2,p3,p4,l1,l2,l3,V1) \par \pard\ri4\plain\f3\fs28\cf0 \par Offensichtlich folgen wir mit dem obigen Verfahren zumindest im \par Anfangswert y(0) = 2 unmittelbar dem eingezeichneten Pfeil. \par \par Nat\'fcrlich k\'f6nnen wir mit MuPAD auch eine Kurve zu einem gegebenen \par Anfangswert mit in die Grafik einzeichnen. Man sieht sehr sch\'f6n, dass \par die von MuPAD berechnete N\'e4herung genauer ist, denn unsere N\'e4herung \par weicht deutlich davon ab. \par \plain\f5\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}f2 := plot::Ode2d( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 (t,Y) -> [f(t, Y[1])], [i/3 $ i=0..9], [2], \par [(t, Y) -> [t, Y[1]], Style = Points, Color = RGB::Red], \par [(t, Y) -> [t, Y[1]], Style = Splines, Color = RGB::Blue]): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}f2 := plot::modify(f2, PointSize = 3*unit::mm): \par {\pntext\f1\'b7\tab}plot(V1, f2, p1,p2,p3,p4,l1,l2,l3, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 Scaling = Constrained, \par XAxisTitle = "t", \par YAxisTitle = "y", \par GridVisible = TRUE, \par XTicksDistance = 1.0, \par YTicksDistance = 0.5):\plain\f4\fs20\cf3 \par \pard\ri4\plain\f3\fs28\cf0 \par Unsere N\'e4herung verbessert sich dann, wenn wir die Schrittweite von \par 1/4 auf kleinere Werte 1/n f\'fcr gro\'dfe n verkleinern. \par \par Wie man sieht, interpoliert MuPAD zwischen den rot eingef\'e4rbten Punkten \par NICHT mit linearen Funktionen, so wie wir es oben gemacht haben. Das \par liegt daran, dass wir die Option \plain\f4\fs32\cf3 Style = Splines\plain\f4\fs20\cf3 \plain\f3\fs28\cf0 verwendet haben. \par Sie bewirkt, dass die N\'e4herungsfunktion wirklich wie eine Kurve aussieht \par und liefert daher offensichtlich viel genauere Werte als unser obiges \par Verfahren. \par \plain\f3\fs28 \par Wir zeichnen noch einige L\'f6sungskurven der betrachteten Differenzial- \par gleichung zum Abschlu\'df: \par \plain\f4\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}[f, t0, Y0] := [numeric::ode2vectorfield( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 \{y'(t) = t*y(t) - y(t)^2, y(0) = 2\}, [y(t)])]: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}G := (t, Y) -> [t, Y[1]]: \par {\pntext\f1\'b7\tab}p1 := plot::Ode2d(f, [-0.5, i $ i = 0..4], Y0, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 [G, Style = Points, Color = RGB::Blue], \par [G, Style = Splines, Color = RGB::Red]): \par \pard\ri4\plain\f4\fs20\cf3 \par \plain\f3\fs28\cf0 W\'e4hlen wir noch 3 und 4 als weitere Anfangswerte. \par \plain\f5\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}[f, t0, Y0] := [numeric::ode2vectorfield( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 \{y'(t) = t*y(t) - y(t)^2, y(0) = 3\}, [y(t)])]: \par G := (t, Y) -> [t, Y[1]]: \par p2 := plot::Ode2d(f, [-0.5, i $ i = 0..5], Y0, \par [G, Style = Points, Color = RGB::Blue], \par [G, Style = Splines, Color = RGB::Green]): \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}[f, t0, Y0] := [numeric::ode2vectorfield( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 \{y'(t) = t*y(t) - y(t)^2, y(0) = 4\}, [y(t)])]: \par G := (t, Y) -> [t, Y[1]]: \par p3 := plot::Ode2d(f, [-0.5, i $ i = 0..5], Y0, \par [G, Style = Points, Color = RGB::Blue], \par [G, Style = Splines, Color = RGB::Black]): \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}plot(p1,p2,p3) \par \pard\ri4\plain\f4\fs22\cf1 \par \plain\f3\fs28\cf0 Wir betrachten noch einige weitere Beispiele aus dem Bereich der Differenzial- \par gleichungen: \par \plain\f7\fs20\cf0 {\pict\wmetafile8\picw4920\pich992\picscalex99\picscaley99\picwgoal2814\pichgoal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}\plain\f7\fs20\cf0 \par \plain\f3\fs28\cf0 Wir zeichnen das Richtungssfeld und eine N\'e4herung zur Anfangsbedingung \par y(0) = 1/2: \par \plain\f7\fs20\cf0 \par \plain\f4\fs24\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}f:= (t, y) -> t*sin(t + y^2) - y \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p1:= plot::VectorField2d([1, f(t, y)], t = 0..4, y = -1..1, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 Mesh = [10, 5], Color = RGB::Black): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p2:= plot::Ode2d( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 (t,Y) -> [f(t, Y[1])], [i/3 $ i=0..12], [0.5], \par [(t, Y) -> [t, Y[1]], Style = Points, Color = RGB::Red], \par [(t, Y) -> [t, Y[1]], Style = Splines, Color = RGB::Blue]): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p2:= plot::modify(p2, PointSize = 3*unit::mm): \par {\pntext\f1\'b7\tab}plot(p1, p2, Scaling = Constrained, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 XAxisTitle = "t", YAxisTitle = "y", \par GridVisible = TRUE, \par XTicksDistance = 1.0, \par YTicksDistance = 0.5):\plain\f4\fs20\cf3 \par \pard\ri4\plain\f7\fs20\cf0 \par \plain\f3\fs28\cf0 Ein letztes Beispiel betrachten wir die Differenzialgleichung \par {\pict\wmetafile8\picw1705\pich1300\picscalex99\picscaley99\picwgoal974\pichgoal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}\plain\f3\fs28\cf0 \par \plain\f4\fs20\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}f := (t, y) -> t*1/y \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p1 := plot::VectorField2d([1, f(t, y)], t = 0..3, y = 0.1..3, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 Mesh = [15, 15], Color = RGB::Black): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p2 := plot::Ode2d( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 (t,Y) -> [f(t, Y[1])], [i/3 $ i=0..9], [0.5], \par [(t, Y) -> [t, Y[1]], Style = Points, Color = RGB::Red], \par [(t, Y) -> [t, Y[1]], Style = Splines, Color = RGB::Blue]): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}p2 := plot::modify(p2, PointSize = 3*unit::mm): \par {\pntext\f1\'b7\tab}plot(p1, p2, Scaling = Constrained, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 XAxisTitle = "t", YAxisTitle = "y", \par GridVisible = TRUE, \par XTicksDistance = 1.0, \par YTicksDistance = 0.5): \par \par \pard\ri4\plain\f3\fs28\cf0 Nun stellt sich die Frage: Ist die Differenzialgleichung, die wir betrachten, \par auch exakt l\'f6sbar? \par \par Eine geschlossene L\'f6sung k\'f6nnen wir mit MuPAD wie folgt berechnen: \par \plain\f4\fs22\cf3 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}eq := ode(y'(x) = t*1/(y(x)), y(x)) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}solve(eq) \par \pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf2\b Anmerkungen:\plain\f3\fs22\cf2 \par \plain\f3\fs20\cf2\b 1\plain\f3\fs20\cf2 . Weitere Anregungen finden Sie in der Buchreihe \plain\f3\fs20\cf3 Mathematik 1 x anders\plain\f3\fs20\cf2 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f8\fs20\cf1 www.schule.mupad.de/literatur\plain\f3\fs20\cf2 kostenfrei kopiert werden. \par \plain\f3\fs20\cf1 \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \par \par }