Math 2650 A. J. Meir Copyright (C) A. J. Meir. All rights reserved. This worksheet is for educational use only. No part of this publication may be reproduced or transmitted for profit in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system without prior written permission from the author. Not for profit distribution of the software is allowed without prior written permission, providing that the worksheet is not modified in any way and full credit to the author is acknowledged.

In order to use Laplace transforms we must load the integral transforms package. restart:with(inttrans): We compute some transforms of functions and some inverse transforms laplace(sinh(t),t,s);

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invlaplace(1/(s^2-1),s,t);

LUklc2luaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0Yn

laplace(t^2*sinh(t),t,s);

LCYqJCwmSSJzRzYiIiIiISIiRichIiRGJyokLCZGJUYnRidGJ0YpRig=

invlaplace(1/((s-1)^3)-1/((s+1)^3),s,t);

KiZJInRHNiIiIiMtSSVzaW5oRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGIyIiIg==

laplace(Heaviside(t-3),t,s);

KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJzR0YoISIkIiIiRishIiI=

invlaplace(exp(-3*s)/s,s,t);

LUkqSGVhdmlzaWRlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJkkidEdGJyIiIiEiJEYr

Now lets look at some discontinuous functions f(t):=piecewise(0<=t and t<1, 1-t, 1<=t, 1);

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMzEiIiFJInRHNiIyRikiIiIsJkYsRixGKSEiIjFGLEYpRiw=

plot(f(t),t=0..5,discont=true);

NigtJSdDVVJWRVNHNiU3UzckJCIjPyEjNSQiKikqKioqKioqKiEiKjckJCIyWiF5JXpnOih6QCEjPSQiMT5fP1IlRz95KiEjOzckJCIybEJHJkhWRndTISM9JCIxd3IvbkRQI2YqISM7NyQkIjFiKyRvJ0g6NGkhIzwkIjElKnBKLlozeiQqISM7NyQkIjEpKT5DMyhwaE4pISM8JCIxLGU8SElRayIqISM7NyQkIjIwOSdRMCMpSFw1ISM8JCIwJ1FoJXosMiYqKSEjOjckJCIyVi5GSDEzdUMiISM8JCIxbEgyUD5mXygpISM7NyQkIjJEJSlSYU1SRFgiISM8JCIxZCxjYTFZWiYpISM7NyQkIjFAekQ2a29rOyEjOyQiMXo/dSllOGBMKSEjOzckJCIydlFYKD1KOnc9ISM8JCIxOFlEIilvJVE3KSEjOzckJCIyR2tlKj5FbiQ0IyEjPCQiMWQ4LyFRRmoheiEjOzckJCIxNmMtOlJFJkcjISM7JCIwUnVcM09acighIzo3JCQiMigpPicqXExdNF0jISM8JCIxLFErbCdcISpcKCEjOzckJCIwIipIVFFBdnIjISM6JCIwNHFlaHhDRyghIzo3JCQiMSIzJkgkb0hpI0ghIzskIjE+XHE7Lnh0cSEjOzckJCIyRmtgYSFmdjpKISM8JCIxZGphJTRXVSlvISM7NyQkIjAmKW9cIjQ3VEwhIzokIjFeNi4mM3opZW0hIzs3JCQiMSY+MCNSWS5LTiEjOyQiMTBbemdgJ3pZJyEjOzckJCIxXE5CJ283VHYkISM7JCIxX2t3OHQpZUMnISM7NyQkIjJhZEl2USpvXVIhIzwkIjFEJXBDaDUkXGchIzs3JCQiMiZSWGUlPWxqOyUhIzwkIjBZOmEiW2pMZSEjOjckJCIxVUlFWSZSPFAlISM7JCIxZHB0YC9FR2MhIzs3JCQiMUgjKltaaC0nZSUhIzskIjFzMl5fUShSVCYhIzs3JCQiMjE1Xy85M0d5JSEjPCQiMSoqeWFmPT48XyEjOzckJCIyKDRrPS9UMSYqXCEjPCQiMGY4ZSplJFwrJiEjOjckJCIxOFgwKFJRYkAmISM7JCIxKFtYSGdoV3klISM7NyQkIjFfLDc8PlkyYSEjOyQiMVspekczUURmJSEjOzckJCIxKm8ybmRXWmgmISM7JCIyOEojSEJhRCZRJSEjPDckJCIxW1dvWXkpKUdlISM7JCIyOmI6TDo3NjwlISM8NyQkIjFma01lX1FRZyEjOyQiMVRObFRaaGhSISM7NyQkIjFoY0d3WjNUaSEjOyQiMidRVnJCXyIqZVAhIzw3JCQiMTNhakohW2hZJyEjOyQiMSNmayRvPiZRYCQhIzs3JCQiMUIpZm1QeCRvbSEjOyQiMXgsTUJFaUpMISM7NyQkIjEqeml1TytWKW8hIzskIjEsc2BLJypwOkohIzs3JCQiMUpvZipvZSp6cSEjOyQiMidvSlM1OC8/SCEjPDckJCIwYUNlIlwnUUgoISM6JCIyJWZhPCUzTmhxIyEjPDckJCIxKHotWStNXlwoISM7JCIxLnNSJipmJ1tdIyEjOzckJCIxRnpVJj49YnEoISM7JCIyRjJzWCE9WyVIIyEjPDckJCIxLyYpbyc+MjcieiEjOyQiMmJcNkwhR3opMyMhIzw3JCQiMUBRdT1YYUUiKSEjOyQiMil5aEQiW2JNKD0hIzw3JCQiMTM0TGAqUlJMKSEjOyQiMkE0cG0vZ2dtIiEjPDckJCIwYSMpKlI8LlkmKSEjOiQiMihmdSxnI29SWCIhIzw3JCQiMWx5YipIbmp2KSEjOyQiMmI4VS9xS09DIiEjPDckJCIxRDk/TVFrXCopISM7JCIxdiYpemxoTl01ISM7NyQkIjEjcFtUYmc2PCohIzskIjEjMzgmZVdSKUcpISM8NyQkIjE6Jio9XHhHcCQqISM7JCIxWFs1M0Q3MmohIzw3JCQiMCh5I0htSzBlKiEjOiQiMkNJQDJQdFk+JSEjPTckJCIxPCI+JyoqNHMjeSohIzskIjJLJCkzUSshenNAISM9NyQkIjEsKyshKSoqKioqKioqISM7JCIxUGhWVioqKioqKj4hI0M3UzckJCIqLCsrKyIhIikkIiM1ISIiNyQkIjEmcFtraSk9KDMiISM6JCIjNSEiIjckJCIxMWRVdjQwajYhIzokIiM1ISIiNyQkIjI4dlg1N20kWzchIzskIiM1ISIiNyQkIjIpXDUkM3pZVUwiISM7JCIjNSEiIjckJCIyXy4leSVHPig+OSEjOyQiIzUhIiI3JCQiMidlIz56QWoqKVwiISM7JCIjNSEiIjckJCIyMVlaNXU6NWUiISM7JCIjNSEiIjckJCIyLSk+XW5YKGVtIiEjOyQiIzUhIiI3JCQiMm8lUWldN1ldPCEjOyQiIzUhIiI3JCQiMjNtUjcwcHUkPSEjOyQiIzUhIiI3JCQiMkVTIlFwYjU5PiEjOyQiIzUhIiI3JCQiMihcISpcUCxRKz8hIzskIiM1ISIiNyQkIjJ4WiNHZCozcTMjISM7JCIjNSEiIjckJCIyLXh0cSg9XHFAISM7JCIjNSEiIjckJCIyMCI0MG1CSVlBISM7JCIjNSEiIjckJCIyRkAjKipwJFtrTCMhIzskIiM1ISIiNyQkIjImKUgsKWZRIkdUIyEjOyQiIzUhIiI3JCQiMSgpZXV5XWssRCEjOiQiIzUhIiI3JCQiMVdFUWZkRiFlIyEjOiQiIzUhIiI3JCQiMU5PTHlnYW1FISM6JCIjNSEiIjckJCIyLkVHSiNlcFtGISM7JCIjNSEiIjckJCIydElaUFk1VyRHISM7JCIjNSEiIjckJCIyYF9dNUVCSiJIISM7JCIjNSEiIjckJCIyRmdybWtEISkqSCEjOyQiIzUhIiI3JCQiMiZHNiZSTzppMyQhIzskIiM1ISIiNyQkIjFRRDRzWilIOyQhIzokIiM1ISIiNyQkIjFBPjBPeSplQyQhIzokIiM1ISIiNyQkIjE3NjxXXmJKTCEjOiQiIzUhIiI3JCQiMlloaCozVE46TSEjOyQiIzUhIiI3JCQiMmIiKmVpIlJWJ1wkISM7JCIjNSEiIjckJCIxX1FgPSNma2UkISM6JCIjNSEiIjckJCIyYyZcbWM0Tm5PISM7JCIjNSEiIjckJCIyJypwOko6P1B2JCEjOyQiIzUhIiI3JCQiMTNuMyNbJCk+JFEhIzokIiM1ISIiNyQkIjJfODFGKGZhPFIhIzskIiM1ISIiNyQkIjEqPlEkM08wKSpSISM6JCIjNSEiIjckJCIxI1sleiVHMkEzJSEjOiQiIzUhIiI3JCQiMUVAVSYpR1trVCEjOiQiIzUhIiI3JCQiMWNNUDl5aF1VISM6JCIjNSEiIjckJCIxRkZMKSlmZExWISM6JCIjNSEiIjckJCIyYmo/SnE3JT1XISM7JCIjNSEiIjckJCIxbnAyRnBhLVghIzokIiM1ISIiNyQkIjFkLjBUdiYpelghIzokIiM1ISIiNyQkIjF0QDtIVVlvWSEjOiQiIzUhIiI3JCQiMXp0SDJeclpaISM6JCIjNSEiIjckJCIxbiU+SDI4QSRbISM6JCIjNSEiIjckJCIxenNyMiUpMzhcISM6JCIjNSEiIjckJCIjXSEiIiQiIzUhIiItJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkIiIhISIiJCIiISEiIi0lJ1BPSU5UU0c2JTckJCIiISEiIiQiIzUhIiI3JCQiIzUhIiIkIiM1ISIiLSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJSVWSUVXRzYkOyQiIiEhIiIkIiNdISIiJShERUZBVUxURy0lKkFYRVNTVFlMRUc2IyUnTk9STUFMRy0lKFNDQUxJTkdHNiMlLlVOQ09OU1RSQUlORURHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkIUghIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlcT4hIiItJS5CT1VORFNfSEVJR0hURzYjJCIkUyohIiItJSlDSElMRFJFTkc2Ig==

laplace(f(t),t,s);

LCYqJEkic0c2IiEiIiIiIiomRiQhIiMsJkYmRicqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRGJEYmRicsJkYkRidGJ0YnRidGJ0YnRic=

This is not too good! so lets try to help Maple along. g(t):=(1-t)*(Heaviside(t)-Heaviside(t-1))+Heaviside(t-1);

LCYqJiwmIiIiRiVJInRHNiIhIiJGJSwmLUkqSGVhdmlzaWRlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNGJkYlLUYrNiMsJkYoRiVGJkYlRihGJUYlRjBGJQ==

plot(g(t),t=0..5,discont=true);

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

laplace(g(t),t,s);

LCYqJEkic0c2IiEiIiIiIiomRiQhIiMsJkYmRicqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRGJEYmRicsJkYkRidGJ0YnRidGJ0YnRic=

h(t):=Heaviside(sin(2*t));

LUkqSGVhdmlzaWRlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSSRzaW5HRiQ2IywkSSJ0R0YnIiIj

plot(h(t),t=0..12,discont=true);

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laplace(h(t),t,s);

LUkobGFwbGFjZUc2IjYlLUkqSGVhdmlzaWRlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSSRzaW5HRig2IywkSSJ0R0YkIiIjRjBJInNHRiQ=

1/(1-exp(-Pi*s))*int(exp(-s*t),t=0..Pi/2);

LCQqKCwmIiIiRiUtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiZJI1BpR0YpRiVJInNHRitGJSEiIkYxRjEsJkYxRiUtRic2IywkRi4jRjEiIiNGJUYlRjBGMUYx

invlaplace(-(exp(-1/2*Pi*s)-1)/((1-exp(-Pi*s))*s),s,t);

LCYjIiIiIiIjRiQpISIiLUkmZmxvb3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiZJInRHRi1GJEkjUGlHRitGJ0YlRiM=

Consider the differential equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1JJm1mcmFjR0YkNigtRiM2Jy1JJW1zdXBHRiQ2JS1GLDYlUSJkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0ZAUSdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zqbi1GLDYlUSJ4RidGPEY/LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZGLUYjNidGKy1GIzYlLUY3NiUtRiw2JVEjZHRGJ0Y8Rj9GQkZIRmBvRkZGK0Zgb0ZGLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZhcC8lKWJldmVsbGVkR0ZRLUZMNi1RIitGJ0ZGRk9GUkZURlZGWEZaRmZuL0ZpblEsMC4yMjIyMjIyZW1GJy9GXG9GanBGQkZgb0ZGRitGYG9GRg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUYjNictSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y/USdub3JtYWxGJy1GIzYlLUYsNiVRI2R0RidGO0Y+RkFGRC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUC8lKWJldmVsbGVkR1EmZmFsc2VGJy1JI21vR0YkNi1RIitGJ0ZELyUmZmVuY2VHRlUvJSpzZXBhcmF0b3JHRlUvJSlzdHJldGNoeUdGVS8lKnN5bW1ldHJpY0dGVS8lKGxhcmdlb3BHRlUvJS5tb3ZhYmxlbGltaXRzR0ZVLyUnYWNjZW50R0ZVLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGZG8tRiw2JVEieEYnRjtGPkZBRkQtRlc2LVEiPUYnRkRGWkZmbkZobkZqbkZcb0Zeb0Zgby9GY29RLDAuMjc3Nzc3OGVtRicvRmZvRl5wLUYjNictRiw2JVEiZkYnRjtGPi1GVzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZERlpGZm5GaG5Gam5GXG9GXm9GYG8vRmNvUSYwLjBlbUYnL0Zmb0ZpcC1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0Y7Rj5GQUZERkRGQUZERitGQUZERitGQUZE 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R0RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0=(0)=0, where 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. We define the right hand side, differential equation, and initial conditions. f(t):=Heaviside(t)-Heaviside(t-2);

LCYtSSpIZWF2aXNpZGVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKCIiIi1GJDYjLCZGKkYrISIjRishIiI=

de:=diff(x(t),t,t)+2*diff(x(t),t)+x(t)=f(t);

LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieEc2IjYjSSJ0R0YsRi5GLiIiIkYoIiIjRipGLywmLUkqSGVhdmlzaWRlRzYkRiZJKF9zeXNsaWJHRixGLUYvLUYzNiMsJkYuRi8hIiNGLyEiIg==

ic:={x(0)=0,D(x)(0)=0};

PCQvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKjYjIiIhRi4vLUYsRi1GLg==

Laplace transform the differential equation. LEq:=laplace(de,t,s);

LywuKiZJInNHNiIiIiMtSShsYXBsYWNlR0YmNiUtSSJ4R0YmNiNJInRHRiZGLkYlIiIiRi8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiw2IyIiISEiIiomRiVGLy1GLEY3Ri9GOSomRiVGL0YoRi9GJ0Y7ISIjRihGLyomRiVGOSwmRi9GLy1JJGV4cEdGMzYjLCRGJUY9RjlGLw==

Substitute the values for the initial conditions and solve the resulting algebraic equation for the transform of the solution. LEqIc:=subs(ic,LEq);

LywoKiZJInNHNiIiIiMtSShsYXBsYWNlR0YmNiUtSSJ4R0YmNiNJInRHRiZGLkYlIiIiRi8qJkYlRi9GKEYvRidGKEYvKiZGJSEiIiwmRi9GLy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCRGJSEiI0YyRi8=

LT:=solve(LEqIc,laplace(x(t),t,s));

LCQqKCwmISIiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkic0dGLCEiI0YmRiZGL0YlLCgqJEYvIiIjRiZGL0YzRiZGJkYlRiU=

Invert to find the solution. invlaplace(LT,s,t);

Error, (in gcd/LinZip) input must be polynomials over the integers

invlaplace(1/(s*(s^2+2*s+1)),s,t)-invlaplace(exp(-2*s)/(s*(s^2+2*s+1)),s,t);

LCgiIiJGIyomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGKiEiIkYjLCZGLUYjRiNGI0YjRi4qJi1JKkhlYXZpc2lkZUdGJzYjLCZGLUYjISIjRiNGIywmRiNGIyomLUYmNiMsJkYtRi4iIiNGI0YjLCZGLkYjRi1GI0YjRi5GI0Yu

sol:=%;

LCgiIiJGIyomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGKiEiIkYjLCZGLUYjRiNGI0YjRi4qJi1JKkhlYXZpc2lkZUdGJzYjLCZGLUYjISIjRiNGIywmRiNGIyomLUYmNiMsJkYtRi4iIiNGI0YjLCZGLkYjRi1GI0YjRi5GI0Yu

And finally graph the solution. plot(sol,t=0..8);

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

Consider the differential equation 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 + 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R0RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0=(0)=0, where 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. We define the right hand side, differential equation, and initial conditions. f(t) := piecewise(0 <= t and t < 4,4-2*t,4 <= t,0);

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMzEiIiFJInRHNiIyRikiIiUsJkYsIiIiRikhIiMxRixGKUYo

Note: Maple can convert automatically to notation using the Heaviside function. f(t):=convert(%,Heaviside);

LCotSSpIZWF2aXNpZGVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKCIiJS1GJDYjLCYhIiUiIiJGKkYwRi8qJkYqRjBGI0YwISIjKiZGKkYwRixGMCIiIw==

de:=diff(x(t),t,t)+4*x(t)=f(t);

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieEc2IjYjSSJ0R0YsRi5GLiIiIkYqIiIlLCotSSpIZWF2aXNpZGVHNiRGJkkoX3N5c2xpYkdGLEYtRjAtRjM2IywmISIlRi9GLkYvRjkqJkYuRi9GMkYvISIjKiZGLkYvRjZGLyIiIw==

ic:={x(0)=1/2,D(x)(0)=0};

PCQvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKjYjIiIhRi4vLUYsRi0jIiIiIiIj

Laplace transform the differential equation. LEq:=laplace(de,t,s);

LywqKiZJInNHNiIiIiMtSShsYXBsYWNlR0YmNiUtSSJ4R0YmNiNJInRHRiZGLkYlIiIiRi8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiw2IyIiISEiIiomRiVGLy1GLEY3Ri9GOUYoIiIlLCYqJEYlRjlGPComRiUhIiMsJkY5Ri8qJi1JJGV4cEdGMzYjLCRGJSEiJUYvLCZGJUYnRi9GL0YvRi9GL0Yn

Substitute the values for the initial conditions and solve the resulting algebraic equation for the transform of the solution. LEqIc:=subs(ic,LEq);

LywoKiZJInNHNiIiIiMtSShsYXBsYWNlR0YmNiUtSSJ4R0YmNiNJInRHRiZGLkYlIiIiRi9GJSMhIiJGJ0YoIiIlLCYqJEYlRjFGMiomRiUhIiMsJkYxRi8qJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCRGJSEiJUYvLCZGJUYnRi9GL0YvRi9GL0Yn

LT:=solve(LEqIc,laplace(x(t),t,s));

Warning, solutions may have been lost

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjTFRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYjNiYtRiw2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y5RlRGV0Y5

LT:=(4/s+(2*(-1+exp(-4*s)*(2*s+1)))/s^2+s/2)/(s^2+4);

KiYsKCokSSJzRzYiISIiIiIlKiZGJSEiIywmRiciIiIqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCRGJSEiJUYsLCZGJSIiI0YsRixGLEYsRixGN0YlI0YsRjdGLCwmKiRGJUY3RixGKEYsRic=

Invert to find the solution. invlaplace(LT,s,t);

Error, (in gcd/LinZip) input must be polynomials over the integers

invlaplace(4/(s*(s^2+4)),s,t)+invlaplace(2*(-1+exp(-4*s)*(2*s+1))/(s^2*(s^2+4)),s,t)+invlaplace(s/(2*(s^2+4)),s,t);

LCwiIiJGIy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiMjISIiRi1GLEYuLUkkc2luR0YmRiojRiMiIiUqJi1JKkhlYXZpc2lkZUdGJjYjLCYhIiVGI0YsRiNGIywqISIpRiMqJC1GMUY3Ri0iIiktRjE2IywmRjtGI0YsRi1GL0YsRi1GI0Yy

sol:=%;

LCwiIiJGIy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiMjISIiRi1GLEYuLUkkc2luR0YmRiojRiMiIiUqJi1JKkhlYXZpc2lkZUdGJjYjLCYhIiVGI0YsRiNGIywqISIpRiMqJC1GMUY3Ri0iIiktRjE2IywmRjtGI0YsRi1GL0YsRi1GI0Yy

And finally graph the solution. plot(sol,t=0..9);

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