\334bung 8 - 15.12.03restart;with(DEtools):# Neben dsolve sind n\374tzliche Funktionen:## dfieldplot(): Richtungsfeld# DEplot() : Richtungsfeld und spezielle L\366sung#

DGL: NiMvLCYtJSVkaWZmRzYkJSJ5Ry0lIiRHNiQlInhHIiIjIiIiKiYlImtHRi1GKEYuRi4iIiE= y := x -> a*sin(k*x)+b*cos(k*x);

NiM+SSJ5RzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJkkiYUdGJSIiIi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiMqJkkia0dGJUYvOSRGL0YvRi8qJkkiYkdGJUYvLUkkY29zR0YyRjVGL0YvRiVGJUYl

y2 := diff(y(x),x$2);

NiM+SSN5Mkc2IiwmKihJImFHRiUiIiItSSRzaW5HNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjKiZJImtHRiVGKUkieEdGJUYpRilGMSIiIyEiIiooSSJiR0YlRiktSSRjb3NHRixGL0YpRjFGM0Y0

y2+k^2*y(x)=0;simplify(%);

NiMvLCgqKEkiYUc2IiIiIi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YnNiMqJkkia0dGJ0YoSSJ4R0YnRihGKEYwIiIjISIiKihJImJHRidGKC1JJGNvc0dGK0YuRihGMEYyRjMqJkYwRjIsJiomRiZGKEYpRihGKComRjVGKEY2RihGKEYoRigiIiE=NiMvIiIhRiQ=

DGL: NiMvLCgtJSVkaWZmRzYkJSJ5Ry0lIiRHNiRGKCIiIyIiIiomIiIkRi0tRiY2JEYoJSJ4R0YtRi0qJiIiJUYtRihGLSEiIiIiIQ== y := x -> a*exp(x)+b*exp(-4*x);

NiM+SSJ5RzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJkkiYUdGJSIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiM5JEYvRi8qJkkiYkdGJUYvLUYxNiMsJEY2ISIlRi9GL0YlRiVGJQ==

y2 := diff(y(x), x$2);

NiM+SSN5Mkc2IiwmKiZJImFHRiUiIiItSSRleHBHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjSSJ4R0YlRilGKSomSSJiR0YlRiktRis2IywkRjAhIiVGKSIjOw==

y1 := diff(y(x), x);

NiM+SSN5MUc2IiwmKiZJImFHRiUiIiItSSRleHBHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjSSJ4R0YlRilGKSomSSJiR0YlRiktRis2IywkRjAhIiVGKUY2

y2+3*y1-4*y(x)=0;

NiMvIiIhRiQ=

restart;with(DEtools):Richtungsfeld:dfieldplot(diff(y(x),x)=1/sqrt(1+x), y(x), x=-2..3, y=-2..3, arrows=MEDIUM);

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L\366sung durch (1,1) mit Richtungsfeld:DEplot(diff(y(x),x)=1/sqrt(1+x), y(x), x=-0.9..3, y=-2..2, [[y(1)=1]]);

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Allgemeine L\366sung:dsolve(diff(y(x),x)=1/sqrt(1+x), y(x));

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Spezielle L\366sung:dsolve({ diff(y(x),x)=1/sqrt(1+x),y(1)=1 }, y(x));

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restart;with(DEtools):

DGL: NiMvLSUlZGlmZkc2JCUieUclInhHKiYsJiomIiIjIiIiRihGLUYtRi0hIiJGLS0lJXNxcnRHNiNGJ0Yt Separation: NiMvKiYlI2R5RyIiIi0lJXNxcnRHNiMlInlHISIiKiYsJiomIiIjRiYlInhHRiZGJkYmRitGJiUjZHhHRiY=Integration: NiMvKiYiIiMiIiItJSVzcXJ0RzYjJSJ5R0YmLCgqJCUieEdGJUYmRi0hIiIlIkNHRiY= => NiMvJSJ5RyomLCgqJCUieEciIiMiIiJGKCEiIiUiQ0dGKkYpIiIlRis= als allgemeine L\366sungAnfangsbedingung: NiMvLSUieUc2IyIiIiIiKg== => NiMvJSJDRyIiJw== und NiMvJSJDRywkIiInISIi => NiMvJiUieUc2IyIiIiomLCgqJCUieEciIiNGJ0YrISIiIiInRidGLCIiJUYt , NiMvJiUieUc2IyIiIyomLCgqJCklInhHRiciIiJGLUYsISIiIiInRi5GJyIiJUYuAllgemeine L\366sung \374berpr\374fen:dsolve(diff(y(x),x)=(2*x-1)*sqrt(y(x)));

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solve(%, y(x));

NiMsLiokSSJ4RzYiIiIlIyIiIkYnKiRGJSIiJCMhIiIiIiMqJkYlRi5JJF9DMUdGJkYpRikqJEYlRi5GKComRiVGKUYwRilGLSokRjBGLkYp

y:=factor(%);

NiM+SSJ5RzYiLCQqJCwoKiRJInhHRiUiIiMiIiJGKiEiIkkkX0MxR0YlRitGKyNGLCIiJQ==

y:=(x)->;

NiMsJCokLCgqJEkieEc2IiIiIyIiIkYnISIiSSRfQzFHRihGKUYpI0YqIiIl

Spezielle L\366sungen:dsolve({diff(y(x),x)=(2*x-1)*sqrt(y(x)), y(1)=9}, y(x));

NiMvLUkieUc2IjYjSSJ4R0YmLUknUm9vdE9mRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiY2IywqKiRJI19aR0YmIyIiIiIiI0Y0KiRGKEY0ISIiRihGMyEiJ0Yz

plot( {(x^2-x+6)^2/4, (x^2-x-6)^2/4}, x=-5..5 );

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

DGL: NiMvLSUlZGlmZkc2JCUieUclInhHKiYsJiIiJCIiIiomRihGLCIiIyEiIkYsRixGJ0YsSeparation: NiMvKiYlI2R5RyIiIiUieUchIiIsJiomJSJ4R0YmIiIjRihGJiIiJEYm Integration: NiMvLSUjbG5HNiMlInlHLCgqJiUieEciIiMiIiUhIiIiIiIqJiIiJEYuRipGLkYuLUYlNiMlIkNHRi4= => NiMvLSUieUc2IyUieEcqJiUiQ0ciIiItJSRleHBHNiMsJiomRiciIiMiIiUhIiJGKiomIiIkRipGJ0YqRipGKg== als allgemeine L\366sungAnfangsbedingung: NiMvLSUieUc2IyIiISomIiIjIiIiLSUkZXhwRzYjRipGKg== => NiMvJSJDRyomIiIjIiIiJSJlR0Yn bzw. NiMvLSUieUc2IyUieEcqKCIiIyIiIi0lJGV4cEc2I0YqRiotRiw2IywmKiZGJ0YpIiIlISIiRioqJiIiJEYqRidGKkYqRio= Allgemeine und spezielle L\366sung:dsolve(diff(y(x),x)=(3+x/2)*y(x));dsolve({diff(y(x),x)=(3+x/2)*y(x), y(0)=2*exp(1)});

restart;with(DEtools):

Gegeben: NiMvLSUlZGlmZkc2JCUieUclInhHKiZGJyIiIiwmKiYiIiNGKkYoRipGKiomRi1GKi0lJXNxcnRHNiNGKEYqISIiRjI= F\374r x muss gelten: x > 0 und NiMwJSJ4RyIiIg==. Die Singularit\344t bei NiMvJSJ4RyIiIg== zeigt sich im Richtungsfeld:dfieldplot(diff(y(x),x)=y(x)/(2*(x-sqrt(x))), y(x), x=0..5, y=-5..5);Separation: NiMvKiYlI2R5RyIiIiUieUchIiIqJiUjZHhHRiYsJiomIiIjRiYlInhHRiZGJiomRi1GJi0lJXNxcnRHNiNGLkYmRihGKA== Die rechte Seite ist kein triviales Integral: Substitution: NiMvJSJ1Ry0lJXNxcnRHNiMlInhH => NiMvKiYlI2R1RyIiIiUjZHhHISIiKiZGJkYmKiYiIiNGJi0lJXNxcnRHNiMlInhHRiZGKA== bzw. NiMvJSNkdUcqJiUjZHhHIiIiKiYiIiNGJy0lJXNxcnRHNiMlInhHRichIiI= Integrand: NiMvKiYlI2R4RyIiIiwmKiYiIiNGJiUieEdGJkYmKiZGKUYmLSUlc3FydEc2I0YqRiYhIiJGLyomRiVGJiooRilGJkYsRiYsJkYsRiZGJkYvRiZGLw== bzw. NiMvJSpJbnRlZ3JhbmRHKiYlI2R1RyIiIiwmLSUlc3FydEc2IyUieEdGJ0YnISIiRi0= bzw. NiMvLSUkSW50RzYkKiYiIiJGKCwmJSJ1R0YoRighIiJGK0YqLSUjbG5HNiNGKQ== R\374cksubstitution und in Separation eingesetzt: NiMvLSUjbG5HNiMtJSRhYnNHNiMlInlHLCYtRiU2IywmLSUlc3FydEc2IyUieEciIiJGMyEiIkYzLUYlNiMlIkNHRjM= => NiMvLSUieUc2IyUieEcqJiUiQ0ciIiIsJi0lJXNxcnRHRiZGKkYqISIiRio= Dies ist die reelle L\366sung f\374r NiMyIiIhJSJ4Rw== und NiMwJSJ4RyIiIg== Maple rechnet das Integral NiMtJSRJbnRHNiQqJiIiIkYnLCYqJiIiI0YnJSJ4R0YnRicqJkYqRictJSVzcXJ0RzYjRitGJyEiIkYwRis= in der komplexen Zahlenebene. Diese Technik f\374hrt zu einem leichtResultat, mit dem Vorteil, dass die Singularit\344t bei NiMvJSJ4RyIiIg== verschwindet:Int(1/(2*x-2*sqrt(x)),x) = int(1/(2*x-2*sqrt(x)),x);In die Separation eingef\374hrt ergibt dies NiMvLSUjbG5HNiMlInlHLCgqJi1GJTYjLCYlInhHIiIiRi4hIiJGLiIiI0YvRi4tJShhcmN0YW5oRzYjLSUlc3FydEc2I0YtRi8tRiU2IyUiQ0dGLg== bzw. NiMvJSJ5RyomJSJDRyIiIi0lJGV4cEc2IywmKiYtJSNsbkc2IywmJSJ4R0YnRichIiJGJyIiI0YyRictJShhcmN0YW5oRzYjLSUlc3FydEc2I0YxRjJGJw==Dies ist offensichtlich eine andere L\366sung:simplify(C*exp(ln(x-1)/2-arctanh(sqrt(x))));Untersucht man den Z\344hler von NiMvLSUieUc2IyUieEcqKiUiQ0ciIiItJSVzcXJ0RzYjLCZGJ0YqRiohIiJGKi1GLDYjLCZGKkYqRidGL0YqLCYtRixGJkYqRipGKkYv , so sieht man: NiMvLSUieUc2IyUieEcqKCUiQ0ciIiItJSVzcXJ0RzYjLCYqJClGJyIiI0YqISIiRipGMkYqLCYtRixGJkYqRipGKkYy , d.h. die Wurzel ist immer komplex, und mit einer komplexen Konstanten entsteht wieder eine relle L\366sung.Der Vorteil dieser L\366sung ist aber, dass NiMtJSJ5RzYjJSJ4Rw== f\374r NiMvJSJ4RyIiIg== nicht mehr beliebig w\344chst.Allgemeine L\366sung:dsolve(diff(y(x),x)=y(x)/(2*x-2*sqrt(x)));Spezielle L\366sung durch (2,1) ("kehrt" \374ber die komplexen Zahlen):dsolve({diff(y(x),x)=y(x)/(2*x-2*sqrt(x)), y(2)=1});

Gegeben: NiMvLSUlZGlmZkc2JCUieUclInhHKiYsJiomIiIjIiIiRihGLUYtKiZGLEYtLSUlc3FydEc2I0YoRi0hIiJGLUYnRjI= , NiMwLSUieUc2IyUieEciIiE= Richtungsfeld:dfieldplot(diff(y(x),x)=(2*(x-sqrt(x))/y(x)), y(x), x=0..6, y=-3..3);Separation: NiMvKiYlInlHIiIiJSNkeUdGJiomLCYqJiIiI0YmJSJ4R0YmRiYqJkYrRiYtJSVzcXJ0RzYjRixGJiEiIkYmJSNkeEdGJg== Integration: NiMvKiQlInlHIiIjLCgqJkYmIiIiKiQlInhHRiZGKUYpKigiIilGKSIiJCEiIilGKyomRi5GKUYmRi9GKUYvKiZGJkYpJSJDR0YpRik= Vergleich:dsolve(diff(y(x),x)=(2*x-2*sqrt(x))/y(x));Kurvenverlauf:DEplot(diff(y(x),x)=(2*(x-sqrt(x))/y(x)), y(x), x=0..6, y=-3..3, [[y(2)=2]]);with(plots, implicitplot):implicitplot(y^2=2*x^2-(8/3)*x^(3/2)+0.5, x=0..3, y=-3..3, numpoints=1000);

restart;with(DEtools):

Gegeben: NiMvLCgtJSVkaWZmRzYkJSJ5RyUieEciIiIqJiIiJEYqRilGKiEiIiomIiIjRipGKEYqRi1GKg== <=> NiMvLSUlZGlmZkc2JCUieUclInhHLCgqJiIiJCIiIkYoRixGLComIiIjRixGJ0YsRixGLEYs Substitution: NiMvJSJ1RywoKiYiIiQiIiIlInhHRihGKComIiIjRiglInlHRihGKEYoRig= => NiMvLSUlZGlmZkc2JCUidUclInhHLCYiIiQiIiIqJiIiI0YrLUYlNiQlInlHRihGK0Yr Neue DGL: NiMvLSUlZGlmZkc2JCUidUclInhHLCYqJiIiIyIiIkYnRixGLCIiJEYs => NiMvKiYlI2R1RyIiIiwmJSJ1R0YmKiYiIiRGJiIiIyEiIkYmRiwqJkYrRiYlI2R4R0Ym => NiMvLSUjbG5HNiMsJiUidUciIiIqJiIiJEYpIiIjISIiRiksJiomRixGKSUieEdGKUYpLUYlNiMlIkNHRik= => NiMvJSJ1RywmKiYlIkNHIiIiLSUkZXhwRzYjKiYiIiNGKCUieEdGKEYoRigqJiIiJEYoRi0hIiJGMQ== R\374cksubstitution: NiMvLCgqJiIiJCIiIiUieEdGJ0YnKiYiIiNGJyUieUdGJ0YnRidGJywmKiYlIkNHRictJSRleHBHNiMqJkYqRidGKEYnRidGJyomRiZGJ0YqISIiRjQ= => NiMvLSUieUc2IyUieEcsKCooJSJDRyIiIiIiIyEiIi0lJGV4cEc2IyomRixGK0YnRitGK0YrKigiIiRGK0YnRitGLEYtRi0qJiIiJkYrIiIlRi1GLQ== Vergleich:dsolve(diff(y(x),x)=3*x+2*y(x)+1);

Gegeben: NiMvLSUlZGlmZkc2JCUieUclInhHLCYtJSRleHBHNiMsJCwmRiciIiJGL0YvISIiRi9GL0Yv Separation: NiMvKiYlI2R5RyIiIiwmLSUkZXhwRzYjLCYlInlHISIiRiZGLUYmRiZGJkYtJSNkeEc=Integral auf der linken Seite ist nicht-trivial:int(1/(exp(-y-1)+1),y);Umformen: NiMvLCYtJSNsbkc2Iy0lJGV4cEc2IywmJSJ5RyEiIiIiIkYtRi0tRiY2IywmRihGLkYuRi5GLi1GJjYjLCZGLkYuLUYpNiMsJkYsRi5GLkYuRi4= Resultat Integration: NiMvLSUjbG5HNiMsJiIiIkYoLSUkZXhwRzYjLCYlInlHRihGKEYoRigsJiUieEdGKC1GJTYjJSJDR0Yo bzw. NiMvLCYiIiJGJS0lJGV4cEc2IywmJSJ5R0YlRiVGJUYlKiYlIkNHRiUtRic2IyUieEdGJQ== Aul\366sen nach y: NiMvLSUkZXhwRzYjLCYlInlHIiIiRilGKSwmKiYlIkNHRiktRiU2IyUieEdGKUYpRikhIiI= => NiMvLSUieUc2IyUieEcsJi0lI2xuRzYjLCYqJiUiQ0ciIiItJSRleHBHRiZGL0YvRi8hIiJGL0YvRjI= Maple:dsolve(diff(y(x),x)=exp(-(y(x)+1))+1, useInt);dsolve(diff(y(x),x)=exp(-(y(x)+1))+1);

Gegeben: NiMvLSUlZGlmZkc2JCUieUclInhHKiYsJiokRigiIiMiIiIqJEYnRixGLUYtKiZGKEYtRidGLSEiIg== Umformen: NiMvLSUlZGlmZkc2JCUieUclInhHLCYqJkYoIiIiRichIiJGKyomRidGK0YoRixGKw== Substitution: NiMvJSJ1RyomJSJ5RyIiIiUieEchIiI= => NiMvLSUlZGlmZkc2JCUieUclInhHLCYqJkYoIiIiLUYlNiQlInVHRihGK0YrRi5GKw== Neue DGL: NiMvLCYqJiUieEciIiItJSVkaWZmRzYkJSJ1R0YmRidGJ0YrRicsJiomRidGJ0YrISIiRidGK0Yn bzw. NiMvKiYlInVHIiIiLSUlZGlmZkc2JEYlJSJ4R0YmKiZGJkYmRiohIiI= Integration: NiMvKiYlInVHIiIjRiYhIiIsJi0lI2xuRzYjJSJ4RyIiIiUiQ0dGLQ== R\374cksubstitution: NiMvKiYlInlHIiIjKiZGJiIiIiokJSJ4R0YmRighIiIsJi0lI2xuRzYjRipGKCUiQ0dGKA== => NiMvKiQlInlHIiIjLCYqKEYmIiIiKiQlInhHRiZGKS0lI2xuRzYjRitGKUYpKihGJkYpRipGKSUiQ0dGKUYp Resultat: NiMvLSYlInlHNiMiIiI2IyUieEcqJkYqRigtJSVzcXJ0RzYjLCYqJiIiI0YoLSUjbG5HRilGKEYoJiUiQ0dGJ0YoRig= , NiMvLSYlInlHNiMiIiM2IyUieEcsJComRioiIiItJSVzcXJ0RzYjLCYqJkYoRi0tJSNsbkdGKUYtRi0mJSJDRzYjRi1GLUYtISIi dsolve(diff(y(x),x)=(x^2+y(x)^2)/(x*y(x)));

Gegeben: NiMvKiglInhHIiIiJSJ5R0YmLSUlZGlmZkc2JEYnRiVGJiomLCZGJkYmKiRGJyIiI0YmRiYsJkYmRiYqJEYlRi5GJiEiIg== Separation: NiMvKiglInlHIiIiLSUlZGlmZkc2JEYlJSJ4R0YmLCZGJkYmKiRGJSIiI0YmISIiKiZGJkYmKiZGKkYmLCZGJkYmKiRGKkYtRiZGJkYu bzw. NiMvKiglInlHIiIiJSNkeUdGJiwmRiZGJiokRiUiIiNGJiEiIiomJSNkeEdGJiomJSJ4R0YmLCZGJkYmKiRGL0YqRiZGJkYrIntegration:int(y/(1+y^2),y);int(1/(x*(1+x^2)), x);Also: NiMvKiYtJSNsbkc2IywmIiIiRikqJCUieUciIiNGKUYpRiwhIiIsKC1GJjYjJSJ4R0YpKiYtRiY2IywmRilGKSokRjFGLEYpRilGLEYtRi0lIkNHRik= bzw. NiMvLSUjbG5HNiMsJiIiIkYoKiQlInlHIiIjRigsKComRitGKC1GJTYjJSJ4R0YoRigtRiU2IywmRihGKCokRjBGK0YoISIiKiZGK0YoJSJDR0YoRig= Somit: NiMvLCYiIiJGJSokJSJ5RyIiI0YlLSUkZXhwRzYjLCgqJkYoRiUtJSNsbkc2IyUieEdGJUYlLUYvNiMsJkYlRiUqJEYxRihGJSEiIiomRihGJS1GLzYjJSJDR0YlRiU= NiMvLSYlInlHNiMiIiI2IyUieEctJSVzcXJ0RzYjLCYtJSRleHBHNiMsKComIiIjRigtJSNsbkdGKUYoRigtRjY2IywmRihGKCokRipGNEYoISIiKiZGNEYoLUY2NiMlIkNHRihGKEYoRihGOw== und NiMvLSYlInlHNiMiIiM2IyUieEcsJC0lJXNxcnRHNiMsJi0lJGV4cEc2IywoKiZGKCIiIi0lI2xuR0YpRjVGNS1GNzYjLCZGNUY1KiRGKkYoRjUhIiIqJkYoRjUtRjc2IyUiQ0dGNUY1RjVGNUY8Rjw= Maple rechnet etwas anders:dsolve(x*y(x)*diff(y(x),x)=(1+y(x)^2)/(1+x^2));Die L\366sungen sind identisch, sie unterscheiden sich in der Darstellung:Sei NiMvLSUieUc2IyIiIkYn => NiMvJiUiQ0c2IyIiIiIiIw== und NiMvJiUiQ0c2IyUiTUciIiQ=# Handrechnung:solve( sqrt(exp(2*ln(1)-ln(2)+2*ln(C1))-1)=1, C1);# Maple-Resultat:solve(sqrt(2*(CM-1))/2=1, CM);Vergleich:C1 := 2: CM := 3:yh := x -> sqrt(exp(2*ln(x)-ln(1+x^2)+2*ln(C1))-1): ym := x -> 1/(1+x^2)*sqrt((1+x^2)*(CM*x^2-1)):plot({ yh(x), ym(x)}, x=0..5,y=0..2, style=[LINE,POINT]);Die Gleichheit der L\366sungen bzw. das Verh\344ltnis der Konstanten ergibt sich auch aus einem formalen Vergleich:solve(1+y^2 = exp(2*ln(x)-ln(1+x^2)+2*ln(C)), y);Nun k\366nnen die Wurzelausdr\374cke gleichgesetzt werden und nach c1 aufgel\366st werden:solve((1+x^2)*(-1-x^2+c1^2*x^2) = (1+x^2)*(cm*x^2-1), c1);