{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Fo nt 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 128 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 11 0 128 128 1 2 1 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 0 "" 0 "" {TEXT -1 23 "Differenzialgleichungen" } }{PARA 0 "" 0 "" {TEXT -1 138 "In vielen Wissenschaften spielen Differ enzialgleichungen eine wichtige Rolle. Wir wollen uns zun\344chst mit \+ Wachstumsprozessen besch\344ftigen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "1. Exponentielles Wachstum" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " Dabei nimmt man an, \+ dass die Zuwachsrate proportional ist zum augenblicklichen Bestand:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dgl:= D (y)(t) = alpha * y(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d solve( dgl, y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 " Als L \366sungen bekommen wir eine einparametrige Kurvenschar. Eine L\366sun g wird eindeutug festgelegt, wenn wir einen Anfangswert vorgeben:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z:=dsolve( [dgl,y(0)=1], y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=subs(alpha=0.5,rhs(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(f,t=0..5);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 77 "Dieses Wachstum bringt also Exponentialfunktionen hervo r. Man kann sich auch " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "variabl e Wachstumsraten alpha = alpha(t) oder Zerfallsraten (negative alpha) \+ vorstellen." }}{PARA 0 "" 0 "" {TEXT -1 24 "2. Logistisches Wachstum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Modelle dieser Art ber\374cksichtigen, dass ein permanentes ex ponentielles Wachstum unrealistisch ist. Von einem belgischen " }} {PARA 0 "" 0 "" {TEXT -1 53 "Mathematiker namens Verhulst stammt die f olgende Idee" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dgl:= D(y)(t) = alpha * y(t) - beta * y(t)^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "(Dabei stellt man sich beta eher klein vo r. Der Term soll an Einfluss gewinnen, wenn y(t) gro\337 ist." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve( dgl, y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z:=dsolve( \+ [dgl,y(0)=1], y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:= subs(\{alpha=0.5,beta=0.01\},rhs(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(f,t=0..50);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "3. Verallgemeinertes logistisches Wachstum" }}{PARA 0 "" 0 "" {TEXT -1 112 " Hierbei verallgemeinert man alpha zu einem alpha(t), evt. beta zu beta(t) und y(t)^(gamma) statt y(t)^2,z.B." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dgl:= D(y)(t) = 2*y(t) - 0.01* y(t) ^4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve( dgl, y(t)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z:=dsolve( [dgl,y(0)=1] , y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(rhs(z),t=0 ..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "4. Differenzialgleichungen zweiter Ordnung" }}{PARA 0 "" 0 "" {TEXT -1 91 " Die wohl ber\374hmteste D ifferenzialgleichung zweiter Ordnung ist die Schwingungsgleichung" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "dgl:= diff(y(t),t$2) + r*diff(y(t), t) + d*y(t) = f(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 " Dabei beschreibt dies ein schwingungsf\344higes System (meachanisch,elektri sch o.\344.). r fungiert als Reibungskonstante," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 " D hat mit R \374ckstellkraft zu tun und f(t) beschreibt eine von au\337en wirkende Kraft. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "r:=0; d:=4; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "dgl;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dgl 1:=subs(f(t)=0,dgl);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "z1: =dsolve( \{dgl1, y(0)=1, D(y)(0) = 2\}, y(t) );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "plot(rhs(z1),t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dgl2:=subs(f(t)=1,dgl);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "z2:=dsolve( \{dgl2, y(0)=1, D(y)(0) = 2\}, y( t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(z2),t=0.. 6*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dgl3:=subs(f(t)=s in(t),dgl);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "z3:=dsolve( \+ \{dgl3, y(0)=1, D(y)(0) = 2\}, y(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(z3),t=0..6*Pi);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "dgl4:=subs(f(t)=sin(2*t),dgl);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "z4:=dsolve( \{dgl4, y(0)=1, D(y)(0) = 2\}, y (t) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(z4),t=0..6*Pi);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "alias(sigma=Heaviside):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "dgl5:=subs(f(t)=5-5*sigma(t-Pi),dgl);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "z5:=dsolve( \{dgl5, y(0)=1, \+ D(y)(0) = 2\}, y(t) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(z5),t=0..3*Pi);" }} }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "47" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }