DO #26. We use an alias to make the Maple output shorter. alias(Y(s)=laplace(y(t),t,s)): with(inttrans): We enter the differential equation and initial values. (Note the use of Maple's composition operator @ to enter the initial value of the second derivative.) diffeq:=diff(y(t),t$3)-diff(y(t),t$2)+diff(y(t),t)-y(t)=0; inits:=y(0)=1,D(y)(0)=1,(D@@2)(y)(0)=3;

LywqLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUYlNiQtSSJ5RzYiNiNJInRHRi5GMEYwRjAiIiJGKCEiIkYqRjFGLEYyIiIh

NiUvLUkieUc2IjYjIiIhIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRikvLS0tSSNAQEdGLjYkRi0iIiNGMUYnIiIk

laplace(diffeq,t,s); subs(inits,%); expand(%); algeqn:=collect(%,Y(s));

Lyw2KiZJInNHNiIiIiQtSSJZR0YmNiNGJSIiIkYrLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkSSJER0YwIiIjNiNJInlHRiY2IyIiISEiIiomRiVGKy0tRjRGNkY4RitGOiomRiVGNS1GN0Y4RitGOiomRiVGNUYoRitGOkY8RisqJkYlRitGP0YrRisqJkYlRitGKEYrRitGP0Y6RihGOkY5

LywuKiZJInNHNiIiIiQtSSJZR0YmNiNGJSIiIkYrISIkRisqJEYlIiIjISIiKiZGJUYuRihGK0YvKiZGJUYrRihGK0YrRihGLyIiIQ==

LywuKiZJInNHNiIiIiQtSSJZR0YmNiNGJSIiIkYrISIkRisqJEYlIiIjISIiKiZGJUYuRihGK0YvKiZGJUYrRihGK0YrRihGLyIiIQ==

LywoKiYsKiokSSJzRzYiIiIkIiIiKiRGJyIiIyEiIkYnRipGLUYqRiotSSJZR0YoNiNGJ0YqRiohIiRGKkYrRi0iIiE=

Note that the differential equation has been transformed into an algebraic equation. We can then use algebra or Maple's solve command to solve for Y(s). solve(algeqn,Y(s));

KiYsJiIiJCIiIiokSSJzRzYiIiIjRiVGJSwqKiRGJ0YkRiVGJiEiIkYnRiVGLEYlRiw=

Finally, we take the inverse Laplace transform of Y(s). sol:=y(t)=invlaplace(%,s,t);

Ly1JInlHNiI2I0kidEdGJSwoLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiYhIiItSSRzaW5HRitGJkYuLUkkZXhwR0YrRiYiIiM=

Checking: subs(sol,diffeq): expand(%);

LyIiIUYj

subs(t=0,rhs(sol)): simplify(%);

IiIi

diff(rhs(sol),t): subs(t=0,%): simplify(%);

IiIi

diff(rhs(sol),t$2): subs(t=0,%): simplify(%);

IiIk

DO #36 restart: alias(Y(s)=laplace(y(t),t,s)): with(inttrans): diffeq:=t*diff(y(t),t$2)-2*diff(y(t),t)+t*y(t)=0; inits:=y(0)=1,D(y)(0)=0;

LywoKiZJInRHNiIiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYpNiQtSSJ5R0YmNiNGJUYlRiVGJ0YnRiwhIiMqJkYlRidGLkYnRiciIiE=

NiQvLUkieUc2IjYjIiIhIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRig=

laplace(diff(y(t),t$2),t,s); expand(%); l1:=-diff(%,s);

LCgqJkkic0c2IiIiIy1JIllHRiU2I0YkIiIiRiotLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJ5R0YlNiMiIiEhIiIqJkYkRiotRjJGM0YqRjU=

LCgqJkkic0c2IiIiIy1JIllHRiU2I0YkIiIiRiotLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJ5R0YlNiMiIiEhIiIqJkYkRiotRjJGM0YqRjU=

LCgqJkkic0c2IiIiIi1JIllHRiU2I0YkRiYhIiMqJkYkIiIjLUklZGlmZkclKnByb3RlY3RlZEc2JEYnRiRGJiEiIi1JInlHRiU2IyIiIUYm

l2:=laplace(diff(y(t),t),t,s); laplace(y(t),t,s);

LCYqJkkic0c2IiIiIi1JIllHRiU2I0YkRiZGJi1JInlHRiU2IyIiISEiIg==

LUkiWUc2IjYjSSJzR0Yk

l3:=-diff(%,s); subs(inits,l1-2*l2+l3)=0; expand(%);

LCQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiWUc2IjYjSSJzR0YpRishIiI=

LywqKiZJInNHNiIiIiItSSJZR0YmNiNGJUYnISIlKiZGJSIiIy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKEYlRichIiIiIiRGJ0YuRjIiIiE=

LywqKiZJInNHNiIiIiItSSJZR0YmNiNGJUYnISIlKiZGJSIiIy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGKEYlRichIiIiIiRGJ0YuRjIiIiE=

newdiffeq:=collect(%,diff(Y(s),s));

LywoKiYsJiokSSJzRzYiIiIjISIiRioiIiJGKy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJZR0YoNiNGJ0YnRitGKyomRidGK0YwRishIiUiIiRGKyIiIQ==

Note that the differential equation has been transformed into a new differential equation. We can then use Maple's dsolve command to solve for Y(s). (Since Y(s) is an alias, rather than a real function, Maple will not work correctly if we try to write a differential equation viewing Y(s) as a function of s, so we rewrite the equation in terms of Z(s).) (-s^2-1)*diff(Z(s),s)-4*s*Z(s)+3=0; dsolve((-s^2-1)*diff(Z(s),s)-4*s*Z(s)+3=0,Z(s));

LywoKiYsJiokSSJzRzYiIiIjISIiRioiIiJGKy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJaR0YoNiNGJ0YnRitGKyomRidGK0YwRishIiUiIiRGKyIiIQ==

Ly1JIlpHNiI2I0kic0dGJSomLCgqJEYnIiIkIiIiRidGK0kkX0MxR0YlRixGLCwmKiRGJyIiI0YsRixGLCEiIw==

Finally, we take the inverse Laplace transform of Y(s). convert(rhs(%),parfrac,s); Z(s):=expand(%);

LCYqJiwmSSRfQzFHNiIiIiJJInNHRiYiIiNGJywmKiRGKEYpRidGJ0YnISIjRicqJkYoRidGKiEiIkYn

LCgqJiwmKiRJInNHNiIiIiMiIiJGKUYpISIjSSRfQzFHRidGKUYpKiZGJEYqRiZGKUYoKiZGJkYpRiQhIiJGKQ==

Since the first term is the derivative of 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 , which is the Laplace transform of 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 , the inverse Laplace transform of the first term is 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 . By the hint in the book, the inverse Laplace transform of the second term is 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 , and the inverse Laplace transform of 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 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 . Hence the solution is sol:=y(t)=t*sin(t)+_C1/2*(sin(t)-t*cos(t))+cos(t);

Ly1JInlHNiI2I0kidEdGJSwoKiZGJyIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRipGKiomSSRfQzFHRiVGKiwmRitGKiomRidGKi1JJGNvc0dGLUYmRiohIiJGKiNGKiIiI0Y0Rio=

Checking: subs(sol,diffeq): expand(%);

LyIiIUYj

subs(t=0,rhs(sol)): simplify(%);

IiIi

diff(rhs(sol),t): subs(t=0,%): simplify(%);

IiIh

DO #14. restart: f:=Int((t-v)*exp(3*v),v=0..t);

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJiwmSSJ0R0YnIiIiSSJ2R0YnISIiRiwtSSRleHBHRiQ2IywkRi0iIiRGLC9GLTsiIiFGKw==

This integral is the convolution of t and exp(3*t). Hence the Laplace transform is the product of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0LUYjNiYtSSNtaUdGJDYjUSFGJy1GIzYkLUklbXN1cEdGJDYlLUY6NiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNVEnaXRhbGljRictRjE2JFEiMkYnRjQvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjRGOUY0LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZVLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjQ= and 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 . 1/s^2*1/(s-3);

KiZJInNHNiIhIiMsJkYjIiIiISIkRichIiI=

We can compare this result to the output of Maple's laplace command. with(inttrans): laplace(f,t,s);

KiZJInNHNiIhIiMsJkYjIiIiISIkRichIiI=

DO #24. restart: with(inttrans): diffeq:=diff(y(t),t$2)+9*y(t)=g(t); inits1:=y(0)=2,D(y)(0)=-3; inits2:=y(0)=0,D(y)(0)=0;

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ0R0YsRi5GLiIiIkYqIiIqLUkiZ0dGLEYt

NiQvLUkieUc2IjYjIiIhIiIjLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnISIk

NiQvLUkieUc2IjYjIiIhRigvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRidGKA==

alias(Y(s)=laplace(y(t),t,s),G(s)=laplace(g(t),t,s)): The transfer function is calculated as the ratio of Y(s)/G(s) under the assumption that the initial conditions are all 0. laplace(diffeq,t,s); subs(inits2,%); expand(%); collect(%,Y(s));

LywqKiZJInNHNiIiIiMtSSJZR0YmNiNGJSIiIkYrLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0kieUdGJjYjIiIhISIiKiZGJUYrLUYzRjRGK0Y2RigiIiotSSJHR0YmRio=

LywmKiZJInNHNiIiIiMtSSJZR0YmNiNGJSIiIkYrRigiIiotSSJHR0YmRio=

LywmKiZJInNHNiIiIiMtSSJZR0YmNiNGJSIiIkYrRigiIiotSSJHR0YmRio=

LyomLCYqJEkic0c2IiIiIyIiIiIiKkYpRiktSSJZR0YnNiNGJkYpLUkiR0dGJ0Yt

Hence, the transfer function H(s) is given by solve(%,Y(s))/G(s): H(s):=expand(%);

KiQsJiokSSJzRzYiIiIjIiIiIiIqRighIiI=

The impulse response function h(t) is the inverse Laplace transform of the transfer function. h(t):=invlaplace(H(s),s,t);

LCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIyIiIkYs

We add the convolution of g(t) and the impulse response function to a particular solution of the homogeneous initial value problem having non-zero initial values to get the solution to the nonhomogeneous problem with the same initial values but with right hand side g(t).(The point of this maneuver is to separate the effects of the non-zero initial values from those of the non-zero input function g.) yp:=dsolve({lhs(diffeq)=0,inits1},y(t));

Ly1JInlHNiI2I0kidEdGJSwmLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEYnIiIkISIiLUkkY29zR0YrRi4iIiM=

nivsol:=Int(subs(t=t-v,h(t))*subs(t=v,g(t)),v=0..t)+rhs(yp);

LCgtSSRJbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwkKiYtSSRzaW5HRiU2IywmSSJ0R0YoIiIkSSJ2R0YoISIkIiIiLUkiZ0dGKDYjRjJGNCNGNEYxL0YyOyIiIUYwRjQtRi02IywkRjBGMSEiIi1JJGNvc0dGJUY9IiIj

We can compare our result to that of dsolve with the laplace option. dsolve({diffeq,inits1},y(t),method=laplace);

Ly1JInlHNiI2I0kidEdGJSwoLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEYnIiIkISIiLUkkY29zR0YrRi4iIiMtSSRpbnRHRis2JCwkKiYtSSJnR0YlNiNJJF9VMUdGJSIiIi1GKjYjLCZGJ0YwRj0hIiRGPkYxL0Y9OyIiIUYnI0YxRjA=

DO #22. restart: with(inttrans): diffeq:=diff(y(t),t$2)+y(t)=Dirac(t-2*Pi); inits:=y(0)=0,D(y)(0)=1;

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ0R0YsRi5GLiIiIkYqRi8tSSZEaXJhY0c2JEYmSShfc3lzbGliR0YsNiMsJkYuRi9JI1BpR0YmISIj

NiQvLUkieUc2IjYjIiIhRigvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRiciIiI=

alias(Y(s)=laplace(y(t),t,s)): laplace(diffeq,t,s); subs(inits,%); collect(%,Y(s)); solve(%,Y(s)); expand(%); Y(s)=simplify(%,exp);

LywqKiZJInNHNiIiIiMtSSJZR0YmNiNGJSIiIkYrLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0kieUdGJjYjIiIhISIiKiZGJUYrLUYzRjRGK0Y2RihGKy1JJGV4cEdGLzYjLCQqJkYlRitJI1BpR0YwRishIiM=

LywoKiZJInNHNiIiIiMtSSJZR0YmNiNGJSIiIkYrISIiRitGKEYrLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMsJComRiVGK0kjUGlHRjBGKyEiIw==

LywmISIiIiIiKiYsJiokSSJzRzYiIiIjRiVGJUYlRiUtSSJZR0YqNiNGKUYlRiUtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2IywkKiZGKUYlSSNQaUdGMkYlISIj

KiYsJiIiIkYkLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComSSJzR0YqRiRJI1BpR0YoRiQhIiNGJEYkLCYqJEYuIiIjRiRGJEYkISIi

LCYqJCwmKiRJInNHNiIiIiMiIiJGKUYpISIiRikqJkYkRiotSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2IyomRiZGKUkjUGlHRi9GKSEiI0Yp

Ly1JIllHNiI2I0kic0dGJSwmKiQsJiokRiciIiMiIiJGLUYtISIiRi0qJkYqRi4tSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZGJ0YtSSNQaUdGM0YtISIjRi1GLQ==

Multiplication by 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 translates f(t)*Heaviside(t) right by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1 , i.e., we get f(t-2*Pi)*Heaviside(t-2*Pi). Hence, the inverse Laplace transform of the second term is sin(t-2*Pi)*Heaviside(t-2*Pi). We can compare this to the output of Maple's dsolve with the laplace option. sol:=dsolve({diffeq,inits},y(t),method=laplace);

Ly1JInlHNiI2I0kidEdGJSomLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiYiIiIsJkYuRi4tSSpIZWF2aXNpZGVHRis2IywmRidGLkkjUGlHRiwhIiNGLkYu

We plot the solution. plot(rhs(sol),t=-5..15);

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