# SECTION 7.5# Example 1restart; de := diff(y(t),t,t) - 2*diff(y(t),t) + 5*y(t) = -8*exp(-t);

NiM+JSNkZUcvLCgtJSVkaWZmRzYkLSUieUc2IyUidEctJSIkRzYkRi0iIiMiIiIqJkYxRjItRig2JEYqRi1GMiEiIiomIiImRjJGKkYyRjIsJComIiIpRjItJSRleHBHNiMsJEYtRjZGMkY2

with(inttrans);

NiM3LyUpYWRkdGFibGVHJShmb3VyaWVyRyUrZm91cmllcmNvc0clK2ZvdXJpZXJzaW5HJSdoYW5rZWxHJShoaWxiZXJ0RyUraW52Zm91cmllckclK2ludmhpbGJlcnRHJStpbnZsYXBsYWNlRyUqaW52bWVsbGluRyUobGFwbGFjZUclJ21lbGxpbkclKnNhdmV0YWJsZUc=

alias(Y = laplace(y(t),t,s));

NiMlIllH

laplace(de,t,s);

NiMvLC4qJiklInNHIiIjIiIiJSJZR0YpRiktLSUiREc2IyUieUc2IyIiISEiIiomRidGKS1GL0YwRilGMiooRihGKUYnRilGKkYpRjIqJkYoRilGNEYpRikqJiIiJkYpRipGKUYpLCQqJiIiKUYpLCZGKUYpRidGKUYyRjI=

subs({y(0)=2,D(y)(0)=12},%);

NiMvLCwqJiklInNHIiIjIiIiJSJZR0YpRikiIikhIiIqJkYoRilGJ0YpRiwqKEYoRilGJ0YpRipGKUYsKiYiIiZGKUYqRilGKSwkKiZGK0YpLCZGKUYpRidGKUYsRiw=

solve(%,Y);

NiMsJCoqIiIjIiIiJSJzR0YmLCYiIiZGJkYnRiZGJiwqKiQpRidGJUYmISIiKiQpRiciIiRGJkYmKiZGMEYmRidGJkYmRilGJkYtRiY=

Ysol := %;

NiM+JSVZc29sRywkKioiIiMiIiIlInNHRigsJiIiJkYoRilGKEYoLCoqJClGKUYnRighIiIqJClGKSIiJEYoRigqJkYyRihGKUYoRihGK0YoRi9GKA==

invlaplace(Ysol,s,t);

NiMsJiomLSUkZXhwRzYjJSJ0RyIiIiwmKiYiIiRGKS0lJGNvc0c2IywkKiYiIiNGKUYoRilGKUYpRikqJiIiJUYpLSUkc2luR0YvRilGKUYpRiktRiY2IywkRighIiJGOg==

# Example 2de := diff(y(t),t,t) + 4*diff(y(t),t) - 5*y(t) = t*exp(t);

NiM+JSNkZUcvLCgtJSVkaWZmRzYkLSUieUc2IyUidEctJSIkRzYkRi0iIiMiIiIqJiIiJUYyLUYoNiRGKkYtRjJGMiomIiImRjJGKkYyISIiKiZGLUYyLSUkZXhwR0YsRjI=

laplace(de,t,s);

NiMvLC4qJiklInNHIiIjIiIiJSJZR0YpRiktLSUiREc2IyUieUc2IyIiISEiIiomRidGKS1GL0YwRilGMiooIiIlRilGJ0YpRipGKUYpKiZGNkYpRjRGKUYyKiYiIiZGKUYqRilGMiomRilGKSokKSwmRidGKUYpRjJGKEYpRjI=

subs({y(0)=1,D(y)(0)=0},%);

NiMvLCwqJiklInNHIiIjIiIiJSJZR0YpRikiIiUhIiJGJ0YsKihGK0YpRidGKUYqRilGKSomIiImRilGKkYpRiwqJkYpRikqJCksJkYnRilGKUYsRihGKUYs

solve(%,Y);

NiMqJiwqKiYiIiMiIiIpJSJzR0YmRidGJyomIiIoRidGKUYnISIiIiImRicqJClGKSIiJEYnRidGJywsKiQpRikiIiVGJ0YnKiZGJkYnRi9GJ0YnKiYiIzdGJ0YoRidGLComIiM5RidGKUYnRidGLUYsRiw=

Ysol := %;

NiM+JSVZc29sRyomLCoqJiIiIyIiIiklInNHRihGKUYpKiYiIihGKUYrRikhIiIiIiZGKSokKUYrIiIkRilGKUYpLCwqJClGKyIiJUYpRikqJkYoRilGMUYpRikqJiIjN0YpRipGKUYuKiYiIzlGKUYrRilGKUYvRi5GLg==

convert(Ysol,parfrac,s);

NiMsKiooIiQiPSIiIiIkOyMhIiIsJiUic0dGJkYmRihGKEYmKiZGJkYmKiYiI09GJilGKSIiI0YmRihGKCooIiNORiZGJ0YoLCYiIiZGJkYqRiZGKEYmKiZGJkYmKiYiIidGJilGKSIiJEYmRihGJg==

invlaplace(%,s,t);

NiMsJiomIyIjTiIkOyMiIiItJSRleHBHNiMsJComIiImRiglInRHRighIiJGKEYoKiYjRihGJ0YoKiYtRio2I0YvRigsKCIkIj1GKComIiInRihGL0YoRjAqJiIjPUYoKUYvIiIjRihGKEYoRihGKA==

# Example 4

de := diff(y(t),t,t) + 2*t*diff(y(t),t) - 4*y(t) = 1;

NiM+JSNkZUcvLCgtJSVkaWZmRzYkLSUieUc2IyUidEctJSIkRzYkRi0iIiMiIiIqKEYxRjJGLUYyLUYoNiRGKkYtRjJGMiomIiIlRjJGKkYyISIiRjI=

laplace(de,t,s);

NiMvLCwqJiklInNHIiIjIiIiJSJZR0YpRiktLSUiREc2IyUieUc2IyIiISEiIiomRidGKS1GL0YwRilGMiomIiInRilGKkYpRjIqKEYoRilGJ0YpLSUlZGlmZkc2JEYqRidGKUYyKiZGKUYpRidGMg==

subs({y(0)=0,D(y)(0)=0},%);

NiMvLCgqJiklInNHIiIjIiIiJSJZR0YpRikqJiIiJ0YpRipGKSEiIiooRihGKUYnRiktJSVkaWZmRzYkRipGJ0YpRi0qJkYpRilGJ0Yt

# This is a first order linear differential equation which can be solved by the#methods of Section 2.3 Y := 'Y';

NiM+JSJZR0Yk

denew := diff(Y(s),s) +(3/s-s/2)*Y(s) = -1/(2*s^2);# Put into standard form

NiM+JSZkZW5ld0cvLCYtJSVkaWZmRzYkLSUiWUc2IyUic0dGLSIiIiomLCYqJiIiJEYuRi0hIiJGLiomIiIjRjNGLUYuRjNGLkYqRi5GLiwkKiZGLkYuKiZGNUYuKUYtRjVGLkYzRjM=

P := (3/s-s/2);

NiM+JSJQRywmKiYiIiQiIiIlInNHISIiRigqJiIiI0YqRilGKEYq

mu := exp(int(P,s));

NiM+JSNtdUctJSRleHBHNiMsJiomIiIkIiIiLSUjbG5HNiMlInNHRitGKyomIiIlISIiRi8iIiNGMg==

simplify(%);

NiMqJiklInNHIiIkIiIiLSUkZXhwRzYjLCQqJiIiJSEiIkYlIiIjRi5GJw==

mu := %;

NiM+JSNtdUcqJiklInNHIiIkIiIiLSUkZXhwRzYjLCQqJiIiJSEiIkYnIiIjRjBGKQ==

denew2 := denew*mu;

NiM+JSdkZW5ldzJHLyooKSUic0ciIiQiIiItJSRleHBHNiMsJComIiIlISIiRigiIiNGMUYqLCYtJSVkaWZmRzYkLSUiWUc2I0YoRihGKiomLCYqJkYpRipGKEYxRioqJkYyRjFGKEYqRjFGKkY3RipGKkYqLCQqJiNGKkYyRioqJkYoRipGK0YqRipGMQ==

# Check that LHS of above equation is the derivative of mu*Y(s):lhs(denew2) - diff(mu*Y(s),s);

NiMsKiooKSUic0ciIiQiIiItJSRleHBHNiMsJComIiIlISIiRiYiIiNGL0YoLCYtJSVkaWZmRzYkLSUiWUc2I0YmRiZGKComLCYqJkYnRihGJkYvRigqJkYwRi9GJkYoRi9GKEY1RihGKEYoRigqKkYnRigpRiZGMEYoRilGKEY1RihGLyomI0YoRjBGKCooKUYmRi5GKEYpRihGNUYoRihGKCooRiVGKEYpRihGMkYoRi8=

simplify(%);

NiMiIiE=

# Solve denew2 by integrating both sides: RHS := rhs(denew2);

NiM+JSRSSFNHLCQqJiMiIiIiIiNGKComJSJzR0YoLSUkZXhwRzYjLCQqJiIiJSEiIkYrRilGMkYoRihGMg==

int(RHS,s);

NiMtJSRleHBHNiMsJComIiIlISIiJSJzRyIiI0Yp

sol := mu*Y = % + C;

NiM+JSRzb2xHLyooKSUic0ciIiQiIiItJSRleHBHNiMsJComIiIlISIiRigiIiNGMUYqJSJZR0YqLCZGK0YqJSJDR0Yq

solve(sol,Y);

NiMqKCwmLSUkZXhwRzYjLCQqJiIiJSEiIiUic0ciIiNGKyIiIiUiQ0dGLkYuRiwhIiRGJUYr

simplify(%);

NiMqKCwmLSUkZXhwRzYjLCQqJiIiJSEiIiUic0ciIiNGKyIiIiUiQ0dGLkYuRiwhIiQtRiY2IywkKiZGKkYrRixGLUYuRi4=

expand(%);

NiMsJiooJSJzRyEiJC0lJGV4cEc2IywkKiYiIiUhIiJGJSIiIyIiIkYvLUYoNiMsJComRixGLUYlRi5GLUYvRi8qKEYlRiZGJ0YvJSJDR0YvRi8=

simplify(%);

NiMqJiwmIiIiRiUqJi0lJGV4cEc2IywkKiYiIiUhIiIlInNHIiIjRiVGJSUiQ0dGJUYlRiVGLiEiJA==

subs(C=0,%);

NiMqJiIiIkYkKiQpJSJzRyIiJEYkISIi

invlaplace(%,s,t);

NiMsJComIiIjISIiJSJ0R0YlIiIi

# Check solution using dsolve:dsolve({de,y(0)=0,D(y)(0) = 0});

NiMvLSUieUc2IyUidEcsJComIiIjISIiRidGKiIiIg==

# Example 5# Since the symbol I is used in this example, and I is Maple's reserved symbol# for sqrt(-1), we begin by defining a new sybol for sqrt(-1), as in Example 1, # Section 3.5interface(imaginaryunit=j);

NiNeIyIiIg==

de := I*diff(y(t),t,t) = -k*(e(t));

NiM+JSNkZUcvKiYlIklHIiIiLSUlZGlmZkc2JC0lInlHNiMlInRHLSUiJEc2JEYvIiIjRigsJComJSJrR0YoLSUiZUdGLkYoISIi

alias(Y=laplace(y(t),t,s),E = laplace(e(t),t,s));

NiQlIllHJSJFRw==

laplace(de,t,s);

NiMvLCYqJiUiSUciIiIsJi0tJSJERzYjJSJ5RzYjIiIhRicqJiUic0dGJy1GLUYuRidGJ0YnISIiKihGJkYnKUYxIiIjRiclIllHRidGJywkKiYlImtHRiclIkVHRidGMw==

subs({y(0)=0,D(y)(0)=0},%);

NiMvKiglIklHIiIiKSUic0ciIiNGJiUiWUdGJiwkKiYlImtHRiYlIkVHRiYhIiI=

subs(Y=E+a/s^2,%);

NiMvKiglIklHIiIiKSUic0ciIiNGJiwmJSJFR0YmKiYlImFHRiZGKCEiI0YmRiYsJComJSJrR0YmRitGJiEiIg==

Esol := solve(%,E);

NiM+JSVFc29sRywkKiglIklHIiIiJSJhR0YoLCYqJkYnRigpJSJzRyIiI0YoRiglImtHRighIiJGMA==

invlaplace(%,s,t);

NiMqKi0lJXNpbmhHNiMqKCwkKiYlIklHIiIiJSJrR0YrISIiI0YrIiIjRipGLSUidEdGK0YrRihGLkYsRi0lImFHRis=

# This unexpected result can be fixed by telling Maple that I>0 and k>0: assume(I>0,k>0);Esol;

NiMsJCooJSNJfGlyRyIiIiUiYUdGJiwmKiZGJUYmKSUic0ciIiNGJkYmJSNrfGlyR0YmISIiRi4=

invlaplace(Esol,s,t);

NiMsJCooKiYlI2t8aXJHISIiJSNJfGlyRyIiIiNGKSIiIyUiYUdGKS0lJHNpbkc2IyomKiZGKEYnRiZGKUYqJSJ0R0YpRilGJw==