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. We will now look at (linearity and at) linear second order differential equations. restart:with(DEtools):with(plots): Consider the equation 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn. This is the prototype linear, second order, differential equation and initial conditions. Together they form an initial value problem (i.v.p.). We consider the differential equation (d.e.): (*) 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 and the corresponding homogeneous d.e. (**) 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 (note the zero r.h.s.). Due to linearity if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtSSNtbkdGJDYkUSIxRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtSSNtbkdGJDYkUSIyRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn are two solutions of the homogeneous d.e. (**) then also so is every linear combination of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtSSNtbkdGJDYkUSIxRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtSSNtbkdGJDYkUSIyRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn, i.e., 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn are arbitrary constants. That is for any constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn, 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 is a solution of (**). Also assume that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtRi82JVEiZ0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= is a solution of (*), (a particular solution), then due to linearity, for any constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn, 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 is a solution of (*). Furthermore if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtRi82JVEiZkYnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= is a solution of (*) with the r.h.s. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGNEY3Rj5GKw== replaced by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGNEY3Rj5GKw==, then (due to linearity) 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 is a solution of (*) with the r.h.s. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGNEY3Rj5GKw== replaced by 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 and also for any constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn, 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 is a solution of (*) with the r.h.s. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGNEY3Rj5GKw== replaced by 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. Examples: Lets define the differential equation: de:=D(D(x))(t)+a*D(x)(t)+b*x(t)=g(t);

LywoLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIjNiNJInhHRis2I0kidEdGKyIiIiomSSJhR0YrRjMtLUYtRi9GMUYzRjMqJkkiYkdGK0YzLUYwRjFGM0YzLUkiZ0dGK0Yx

ic:=x(0)=x_0,D(x)(0)=v_0;

NiQvLUkieEc2IjYjIiIhSSR4XzBHRiYvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRidJJHZfMEdGJg==

We will now consider homogeneous equations (with 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) and with constant coefficients (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== constants). We will choose some values for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and solve the equation using the dsolve command. You can also try solving the equation by hand. g(t):=0;a:=0;b:=9;

IiIh

IiIh

IiIq

sol:=dsolve(de,x(t));

Ly1JInhHNiI2I0kidEdGJSwmKiZJJF9DMUdGJSIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRGJyIiJEYrRisqJkkkX0MyR0YlRistSSRjb3NHRi5GMUYrRis=

g(t):=cos(t);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInRHRic=

sol:=dsolve(de,x(t));

Ly1JInhHNiI2I0kidEdGJSwoKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRiciIiQiIiJJJF9DMkdGJUYyRjIqJi1JJGNvc0dGLEYvRjJJJF9DMUdGJUYyRjItRjZGJiNGMiIiKQ==

Observe the homogeneous solution is 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 and the particular solution is 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. g(t):=0;

IiIh

x_0:=1;v_0:=0;

IiIi

IiIh

de;ic;

LywmLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIjNiNJInhHRis2I0kidEdGKyIiIi1GMEYxIiIqIiIh

NiQvLUkieEc2IjYjIiIhIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRig=

sol1:=dsolve({de,ic},x(t));

Ly1JInhHNiI2I0kidEdGJS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRGJyIiJA==

sol1:=rhs(sol1);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGJyIiJA==

plot(sol1,t=0..4);

NictJSdDVVJWRVNHNiQ3XnI3JCQiIiEhIiIkIiM1ISIiNyQkIjJMTEwkM3gmKSozIiEjPSQiMVpLJD5VYlkqKiohIzs3JCQiMm1tbW1UOih6QCEjPSQiMCRmait1aXkqKiEjOjckJCIrREpkcEshIzYkIjFtLXJvSSQ+JioqISM7NyQkIjFMTExMM1ZmViEjPCQiMU9XOWc0ZzkqKiEjOzckJCIqRFkiUmwhIzUkIjEleTwzTyU+MykqISM7NyQkIjFtbW1tOycpPSgpISM8JCIxJVJ6KFJDJylmJyohIzs3JCQiLEQiej5eNyEjNiQiMV9iJjRMZlBJKiEjOzckJCIyTUxMJGUnNDBqIiEjPCQiMT5fYi5ASkYpKSEjOzckJCIsdlEmM2Q/ISM2JCIxOlhCLUBYYiIpISM7NyQkIjJubW1tNmhPWyMhIzwkIjE4U3AvQkBddCEjOzckJCIybW1tbVlrSSJIISM8JCIxX2p3cyRmIz1rISM7NyQkIjJtbW1tInlZVUwhIzwkIjF5LF8jZlYqemAhIzs3JCQiLHZHSSlwUCEjNiQiMDpkc103IWVVISM6NyQkIjFMTExlRj4oPiUhIzskIjFRcU5UaT1tSSEjOzckJCIyJSoqKioqXFo3TWYlISM8JCIyWnEiM14mM2UiPiEjPDckJCIya21tOz5LJyopXCEjPCQiMU4iKj1iMCVSUSghIzw3JCQiMUtMTGVaKikqUiYhIzskITEpZVsrd0dfIlwhIzw3JCQiMSoqKioqXEtkLCJlISM7JCEyJXkkKSlcVS5TciIhIzw3JCQiMUtMTGU5WE1pISM7JCEwQCFINCgpel1IISM6NyQkIjFtbW0iZlgoZW0hIzskITE5PXYnelspUlQhIzs3JCQiMUxMTDMhejszKCEjOyQhMVk0QiRlVSVlXyEjOzckJCIxKioqKipcVTdZXSghIzskITFjQXdBbVwjSCchIzs3JCQiMW1tO0g5bFJ6ISM7JCExJGZaImVjV11zISM7NyQkIjFNTExML3B1JCkhIzskITFZSmJaMzImMykhIzs3JCQiMW5tbTtjMFQiKiEjOyQhMU4jKXBPcVU4IyohIzs3JCQiMUxMTGUlR0NkKiEjOyQhMXppeT44MlEnKiEjOzckJCIqOCFRKzUhIikkITFZRnVZeV8sKiohIzs3JCQiMioqKioqXEsoMzcsIiEjOyQhMVglXGsiR3hUKiohIzs3JCQiMikqKioqKlxMUD8tIiEjOyQhMURsbyEpb19yKiohIzs3JCQiLHYkZidHLiIhIzUkITE2diM+a2UyKioqISM7NyQkIiphJXBWNSEiKSQhMSczWmgheVcqKioqISM7NyQkIixEOUJYMCIhIzUkITFuZ0IvX2UoKioqISM7NyQkIitYPE5sNSEiKiQhMWtwbixHPCYpKiohIzs3JCQiLHZNIT13NSEjNSQhMUoqcGtwQkEnKiohIzs3JCQiKiYqM3EzIiEiKSQhMTE4SjBAd0cqKiEjOzckJCIyKioqKioqKjQvdkc2ISM7JCExbW54RzM/LSgqISM7NyQkIiooPVxxNiEiKSQhMSZcIikpUm1wQiQqISM7NyQkIjJubW0iZkJJWTchIzskITFqbkFyLVxvIykhIzs3JCQiMkxMTExPW2tMIiEjOyQhMXBpKipcelVsayEjOzckJCIyTUxMJDM2anU4ISM7JCExJ3A0cnMuNmImISM7NyQkIjJLTExMJlEiR1QiISM7JCExVyJIaFVAU2MlISM7NyQkIjJubTtIWUhzWCIhIzskITI4OGxGJ1xaVEwhIzw3JCQiLEQyWDtdIiEjNSQhMj0nKmU7RypvZj8hIzw3JCQiMmxtO0hUZzRhIiEjOyQhMU9DUkVzIylRKikhIzw3JCQiMkpMTEx2di1lIiEjOyQiMFh2JWZZV1ZHISM7NyQkIjJsbTtIIjRUQjshIzskIjI7OHRLUCIqPWQiISM8NyQkIixEMllsbSIhIzUkIjIwNWMpW0M6TEchIzw3JCQiK1g0aTI8ISIqJCIxKFE4OmxRLypSISM7NyQkIix2ImVwWzwhIzUkIjEvZzFWJTRzMyYhIzs3JCQiMm5tO3o4YDp6IiEjOyQiMSVSZClbKikzXGghIzs3JCQiMk1MTCRlL1RNPSEjOyQiMVgpeSFlJGYlNHIhIzs3JCQiMk5MJDNkb3d0PSEjOyQiMVc1MGRFTikpeSEjOzckJCIyTExMZURCSiI+ISM7JCIxZHpYa3RTZCYpISM7NyQkIi12W1dkYj4hIzYkIjEzb0VbKmVfOSohIzs3JCQiMm1tbTtrRCEpKj4hIzskIjFMOiZvYSUpXGUqISM7NyQkIjFuOy9ySTI/PyEjOiQiMTwsaXknZUN2KiEjOzckJCIybG07L10/QC8jISM7JCIxXG8lUkckR3gpKiEjOzckJCIxbTt6SHo7az8hIzokIjBuPSp5QyIqZSoqISM6NyQkIjJqbW0iZmBAJzMjISM7JCIxJlw+QEcqKXAqKiohIzs3JCQiMUwkM19gNmU0IyEjOiQiMWUlXHVyNCoqKioqISM7NyQkIi1ENnhTMEAhIzYkIjFGRlkxSWElKioqISM7NyQkIjJtbSJIKClRKzpAISM7JCIxdDhuJ2YkKjMpKiohIzs3JCQiMkxMTEwxK1k3IyEjOyQiMUhKayp6cyplKiohIzs3JCQiMm9tO2FUI3pWQCEjOyQiMSVvKlxbbFIhKikqISM7NyQkIix2dyUpSDsjISM1JCIxSyhlJkc6Lyp5KiEjOzckJCIyTEwkZSpIVFc/IyEjOyQiMTh0NVtVKCpmJSohIzs3JCQiMmxtbTskeSplQyMhIzskIjE2IkhYX3FaKSopISM7NyQkIio5YjpMIyEiKSQiMSkqNFAiZWlSZCghIzs3JCQiMigqKioqXEFZWHRCISM7JCIxYkBEPzhoJnAnISM7NyQkIjInKioqKipcNWFgVCMhIzskIjEpPVlAJnpnNmQhIzs3JCQiMicqKipcKDNTKmVYIyEjOyQiMjhcVSJlcGR0WSEjPDckJCIyKCoqKipcN1JWJ1wjISM7JCIxLTwmKSlmLWxjJCEjOzckJCIyKCoqKlxQY1k5YSMhIzskIjJtXT9mISkpR3dBISM8NyQkIjInKioqKipcQGZrZSMhIzskIjFzYXYkZSRIWSUqISM8NyQkIjJqbW1UMzBwaSMhIzskITIvXWIhR00vdEUhIz03JCQiMUxMTGA0Tm5FISM6JCEyMFtud1IyYFoiISM8NyQkIjJqbW07Yk4wciMhIzskITJZQ2EmW1FzU0YhIzw3JCQiKjo/UHYjISIpJCExYDNvJnkuLSdSISM7NyQkIjFMTGU5PSZHeiMhIzokITEvTWdsO1czXSEjOzckJCIybG1tInpNKT4kRyEjOyQhMScpeng9WHQoKWYhIzs3JCQiMUxMZUNad3VHISM6JCExUGZqLWVmanAhIzs3JCQiKihmYTxIISIpJCExb2R3LyQ0XCN5ISM7NyQkIjFMTCRlZ2AhKSpIISM6JCExMEolZTIkMygzKiEjOzckJCIybW1tVFdJLC8kISM7JCExbkZxVyc0LmEqISM7NyQkIjIpKioqKlwjRzJBMyQhIzskITEqKkdpLjhzVCkqISM7NyQkIjJMTDNGPXdGNSQhIzskITFXPDZjVUZLKiohIzs3JCQiMmxtO0gzWEw3JCEjOyQhMTNkJSo9Kj5dKSoqISM7NyQkIjFMMy1MJkhPOCQhIzokITF0P3BGbTkoKioqISM7NyQkIjInKipcNyQpUiJSOSQhIzskITEiXCRIOnZ2KioqKiEjOzckJCIyaztIS1YpPmFKISM7JCExKSpvYyc0XUcqKiohIzs3JCQiMUxMTCQpR1trSiEjOiQhMWxXT1k0VncqKiEjOzckJCIuRGNoO2c9JCEjNyQhMG1HKClRQzgiKiohIzo3JCQiMmxtO3pNXXY/JCEjOyQhMSdcTiFSInBbISkqISM7NyQkIjFMJDMtMyUzSEshIzokIS84MDkkNHZsKiEjOTckJCIsRCJ5aF1LISM1JCExKlFYKXknZilwJSohIzs3JCQiMUxMZSoqbzQjSCQhIzokITEqZSNSTTMpeSoqKSEjOzckJCIyb21tbSlmZExMISM7JCExbXUhPSJHdidRKSEjOzckJCIybm1tVE0lKmZQJCEjOyQhMS5INSYpXHRGdyEjOzckJCIybW1tO3E3JT1NISM7JCEwLk94TGhgdSchIzo3JCQiLXY4KXovWSQhIzYkITFCWiIqPW5eaWQhIzs3JCQiMkxMTGUjcGEtTiEjOyQhMVo2QEViLClvJSEjOzckJCIybW07SEItN2EkISM7JCExaCh6bG4iXE1PISM7NyQkIiphZCl6TiEiKSQhMkNXXScpZVhAYCMhIzw3JCQiMW5tOyUpMztDTyEjOiQhMmolUXRBLiJ5QSIhIzw3JCQiMkxMTCRHVVlvTyEjOyQiMWBtLydvUiM9KSohIz03JCQiLHZtKjMzUCEjNSQiMjxSUGloKFIkRyIhIzw3JCQiMmxtbW01Onh1JCEjOyQiMm9bbypbeFxdQyEjPDckJCIxTExlKjNrKip5JCEjOiQiMW5dVy0jeWpsJCEjOzckJCIsRDI4QSRRISM1JCIxeE87MCQpZi5bISM7NyQkIip1XUUoUSEiKSQiMWA3WFInKW9IZSEjOzckJCIsdlMpMzhSISM1JCIxOyFbJD4mKjNxbiEjOzckJCItdi5VYWNSISM2JCIxNDtiL2JScHchIzs3JCQiI1MhIiIkIjFAXEsoZVImUSUpISM7LSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJSVWSUVXRzYkOyQiIiEhIiIkIiNTISIiJShERUZBVUxURy0lKkFYRVNTVFlMRUc2IyUnTk9STUFMRy0lKFNDQUxJTkdHNiMlLlVOQ09OU1RSQUlORURHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkKyUhIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlU0QhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlcUYhIiItJSlDSElMRFJFTkc2Ig==

g(t):=sin(2*t);

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGJyIiIw==

de;ic;

LywmLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIjNiNJInhHRis2I0kidEdGKyIiIi1GMEYxIiIqLUkkc2luR0YoNiMsJEYyRi4=

NiQvLUkieEc2IjYjIiIhIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRig=

sol2:=dsolve({de,ic},x(t));

Ly1JInhHNiI2I0kidEdGJSwoLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEYnIiIkIyEiIyIjOi1JJGNvc0dGK0YuIiIiLUYqNiMsJEYnIiIjI0Y2IiIm

sol2r:=rhs(sol2);

LCgtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIyEiIyIjOi1JJGNvc0dGJUYpIiIiLUYkNiMsJEYrIiIjI0YyIiIm

plot(sol2r,t=0..13);

6'-%'CURVESG6$7cz7$$""!!""$"#5!""7$$"2mmTgx=5x"!#=$"1MCS[v!f)**!#;7$$"1LL3_v.UN!#<$"1)Q!*f(RuV**!#;7$$".D"Gj08`!#9$"10,^s"RP()*!#;7$$"1mm;/^2%3(!#<$"1vBiA8?w(*!#;7$$".DcE6E1"!#8$"14u5GW9+&*!#;7$$"2LLL3-:oT"!#<$"1c)pR$4c>"*!#;7$$"-DJDAD@!#7$"1)>\0qJl1)!#;7$$"2lmm;/IO$G!#<$"0k*yOF7rm!#:7$$"2F$ekG;,]M!#<$"1M))pk<#=B&!#;7$$"2$**\i:KRmS!#<$"2'o$Rav?#QO!#<7$$"1mTg-[x#o%!#;$"2<D:Omg![>!#<7$$"1LLe*Qc"*H&!#;$"1Zx61\d>A!#<7$$"1m"z%\>M#*f!#;$!2Q&\t$pJeo"!#<7$$".v$4v_&o'!#8$!1H]unFXwM!#;7$$"1K3FpIryt!#;$!1`(pr6s22&!#;7$$"1mm;H')*=2)!#;$!1;"p")42!*R'!#;7$$"1nm"H2z'p()!#;$!1zP(GvI#4u!#;7$$"1omm;&fuY*!#;$!1G#G]lO$[!)!#;7$$"1oTgFY!>k*!#;$!1l&R&G%em9)!#;7$$"1o;aQ(\j")*!#;$!1^kMeN#)>#)!#;7$$"1o"z%\[z!***!#;$!1McE"eqwE)!#;7$$"2omTg*R_;5!#;$!1NEd!y5,H)!#;7$$"2oTNr]oR."!#;$!1WLWx$GrG)!#;7$$"2o;H#=IT^5!#;$!1Exd=0ye#)!#;7$$"2o"HKHv&)o5!#;$!1_s5gq>0#)!#;7$$"2om;//-j3"!#;$!1Y-X[,eE")!#;7$$"2M3F>W[d:"!#;$!1N$pYdf=d(!#;7$$"/vVV[>D7!#8$!1w)fAPhUl'!#;7$$"2l"z%\CTYH"!#;$!1nP'oQ$H=a!#;7$$"2JLekk(3k8!#;$!1ZCl/:#G#R!#;7$$"2lm;/Nt%G9!#;$!1.eBo'oeO#!#;7$$"2)**\Pa!fG\"!#;$!1C>dD,J$4(!#<7$$"1LLLeZCd:!#:$"1Y]i+='**z*!#<7$$"2km"Hi/j@;!#;$"2Ctzy'>*Rj#!#<7$$"2(**\PH")H)o"!#;$"1_].xp:QU!#;7$$"1L$ekzl\v"!#:$"1O(**))pbIm&!#;7$$"2imTNYL;#=!#;$"1v5a?IDYo!#;7$$"2(**\iI6I))=!#;$"1l"fhrK_t(!#;7$$"2KLe*G]xA>!#;$"1%4v"zwol!)!#;7$$"2km"HF*[s&>!#;$"1G1a@jE-$)!#;7$$"1L$ek(e[u>!#:$"1^eY'=JVQ)!#;7$$"2(**\iDGs"*>!#;$"1WE*pON=W)!#;7$$"2jm"zu(f*3?!#;$"1N@VDcbu%)!#;7$$"1L$eRs'>E?!#:$"1l/W9wL#[)!#;7$$"2&**\i?X9&4#!#;$"1MmUe$HME)!#;7$$"2km"H<B4k@!#;$"1(['ztGC^w!#;7$$",v**=GB#!#5$"1%[4j:FFn'!#;7$$"1L$3xnX:I#!#:$"0YRn$p)fO&!#:7$$"2lm;zNs-P#!#;$"1;YopDB&y$!#;7$$".D"Q!***QC!#7$"2w!QhgF;(*>!#<7$$"1$3-j(Gp4D!#:$"1"*>*41@-@#!#=7$$"2j;zWr'Q!e#!#;$!2O,C;$3L,?!#<7$$"0DcEb!3^E!#9$!1hUlMj()zR!#;7$$"1LL$3Ru<s#!#:$!1mu@a<R@e!#;7$$"2l;z>cTSy#!#;$!0R[%3^jgs!#:7$$".DJt3j%G!#7$!1pR[]=ds%)!#;7$$"2K$3F/fd3H!#;$!1X!y^_<7T*!#;7$$"2mm;a2V3(H!#;$!2M)yZUj(R+"!#;7$$"1$3-)=@*e+$!#:$!23(o?d=oC5!#;7$$"/v=i6%4/$!#8$!2ZSR())yRM5!#;7$$"2&3-)Qol%eI!#;$!28]sDVq]."!#;7$$"1<Hd0-*f2$!#:$!2E^ey_LH."!#;7$$"2ailss9N4$!#;$!2(GTkQC)z-"!#;7$$"2MLe*[#R56$!#;$!2E7)>c,A?5!#;7$$"1n"HdLP6=$!#:$!1Vwpz/*Qh*!#;7$$",DUN7D$!#5$!1ut`(z?")f)!#;7$$"-vQ8i@L!#6$!1fj%=Uvj=(!#;7$$"+bs+#R$!"*$!1'\![IU.Qa!#;7$$"-DrJRiM!#6$!1iX()yNYCM!#;7$$",v3zF`$!#5$!1&R1P)*z#H7!#;7$$"2(******f*31g$!#;$"1We<)Rs#=(*!#<7$$"2(****\K)Q%oO!#;$"2()*[*el*>pJ!#<7$$"2'*****\qoit$!#;$"1=PB9%)ev_!#;7$$",vd)4/Q!#5$"1N9y)*yw1s!#;7$$"2m"z%**f%plQ!#;$"1u2`Ll<V()!#;7$$"1LeRA1HFR!#:$"2-U82^Q>+"!#;7$$"2%\P%[k'))))R!#;$"2B*Q>$GA#*4"!#;7$$"1m;HnE[]S!#:$"2&*QWeeFG;"!#;7$$"0-)QuNzoS!#9$"2tj,*y&z[<"!#;7$$"1uV["[/r3%!#:$"1r`x?*3P="!#:7$$"1G2e)Q:a5%!#:$"2*335D`G*="!#;7$$"1#3xcHEP7%!#:$"1$o3D*ye">"!#:7$$"2lVtF?P?9%!#;$"27kT"z^g!>"!#;7$$"20zp)4"[.;%!#;$"2of4sBNj="!#;7$$"2b9mp,f'yT!#;$"2`e`ui&yy6!#;7$$"/D1C*pp>%!#8$"2*)prMStz;"!#;7$$"1<zW_N@qU!#:$"1c_@a%oF4"!#:7$$"1LL$3=dMM%!#:$"1KePkDS"p*!#;7$$"1M$e99/bS%!#:$"1O<D].#[I)!#;7$$"1LL3-6bnW!#:$"1$>y?))*R`m!#;7$$"1L$3F1)fHX!#:$"2VjMSEY,z%!#<7$$"1LLLB]k"f%!#:$"2&o<U'z0^x#!#<7$$"1#z>hqKxi%!#:$"2^ujS5?*e:!#<7$$"0D1*)Q?Qm%!#9$"2bB\J-<nE$!#=7$$"2vq#pr!3**p%!#;$!19#yCv093*!#<7$$"1m"zWv&*ft%!#:$!28ER4W%*>8#!#<7$$"1DcEPM3sZ!#:$!2Y3MTh(\JL!#<7$$"1$3_+7r"3[!#:$!1#RC8vRN\%!#;7$$"20aQG!)eU%[!#;$!1<<\"4Tag&!#;7$$"1**\i&[Y.)[!#:$!1$4%f,h4bm!#;7$$"2D3-Q"RBW\!#;$!1;A,M'ygK)!#;7$$"1m"z>M@"3]!#:$!1R-4fpJ5(*!#;7$$"1\i:q(3?2&!#:$!2<L$*H0gh2"!#;7$$"1LLL)>'*e8&!#:$!1.yJ*GaX9"!#:7$$"1"z%\:N%4<&!#:$!27Zw`AWc;"!#;7$$"0DcE$3*f?&!#9$!2(Rv.EQ$[<"!#;7$$"1zpB"\9NA&!#:$!2&o8%o*[%\<"!#;7$$"13x")\"Q5C&!#:$!2FI@&\92s6!#;7$$"1P%)R3=ce_!#:$!2Dq+<(oAm6!#;7$$"1n"zpY&3w_!#:$!1()o/KCVd6!#:7$$"2N3-85!=Y`!#;$!2%yivIIX$4"!#;7$$".DctuiT&!#7$!108Ww*R'f)*!#;7$$"1*\PMS@I[&!#:$!1/YIG:Dw%)!#;7$$"-Dr!o(\b!#6$!1yPSM+X)z'!#;7$$"20]i!RZ^;c!#;$!14nd">kI*[!#;7$$".voShKo&!#7$!/Ta<GBNG!#97$$"1$3_Db/Hv&!#:$!1_T'\)>SEh!#<7$$"1n"H#)pZD#e!#:$"1b]vdf\%f"!#;7$$"0D1R%3>#*e!#9$"1"yI6%*fQp$!#;7$$"1LLe*)R$='f!#:$"1jU)ffn&)f&!#;7$$"1$3_v8)yDg!#:$"1vw3FxR4r!#;7$$"1L3_&GU(*3'!#:$"1G0cN`FR$)!#;7$$"1$e*[Lkp`h!#:$"1f!yfr]xC*!#;7$$"1L$e9e]w@'!#:$"1hcTO,U1)*!#;7$$"2&G2eyj*[B'!#;$"1)=noow[*)*!#;7$$"1CJqv@9_i!#:$"1m>6dicc**!#;7$$"0_DG(zQpi!#9$"1%[S$zfU"***!#;7$$"1;z%*pPj'G'!#:$"18<<1]Y****!#;7$$"18.2n&zQI'!#:$"1L")HkVw!)**!#;7$$"13F>k`7@j!#:$"1O"ochwa$**!#;7$$"1/^Jh6PQj!#:$"1DvDF`#Q')*!#;7$$"2&*\P%ephbj!#;$"/W*4N/hw*!#97$$"1$3Fp9+YU'!#:$"1Me6zX!G7*!#;7$$"1mmTNLe$\'!#:$"1U0OqlR.")!#;7$$"1<a)3[P_c'!#:$"1u*3&*ee'*p'!#;7$$"1mTNE;*oj'!#:$"1k1Yeg#[,&!#;7$$"1;H#=xX&3n!#:$"20=y;BOB8$!#<7$$"1m;H<**>!y'!#:$"2i0B(zW/W6!#<7$$"1*\i:1vD%o!#:$!2&H!)4,4'=)f!#=7$$"1LL$e?]\!p!#:$!1l+!)o)zYG#!#;7$$"1nT5]`Knp!#:$!1mP'\5?Z&Q!#;7$$".vV\+(Hq!#7$!1(\\+4EAD&!#;7$$"2D$ekBUVkr!#;$!1\viQ;[fu!#;7$$"1mm"H&z;*H(!#:$!1<rIHnz*G)!#;7$$"1L$e*y5OQu!#:$!1->=%4ozd(!#;7$$"1*****\?avd(!#:$!1kdN'***oBa!#;7$$"/D1k3kXw!#8$!2ch^(o0VgR!#<7$$".DJ_FPr(!#7$!2-!=#Q$*QQJ#!#<7$$"2&*\(=#=9=y(!#;$!2O*y.wgu#e&!#=7$$"1***\7%3!*\y!#:$"21D\LoNrA"!#<7$$"1\P%[=yd"z!#:$"22yFZ&QA2H!#<7$$"1*\P%Gbl")z!#:$"1)z@*3Y!)oW!#;7$$"0DJ?(G`Z!)!#9$"1x4tW")\Xe!#;7$$"1**\i:-T8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xh:=select(has,sol2r,3*t);

LCYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIyEiIyIjOi1JJGNvc0dGJUYpIiIi

xp:=select(has,sol2r,2*t);

LCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIjIyIiIiIiJg==

odetest(x(t)=cos(3*t)-2/15*sin(3*t),de);

LCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIjISIi

odetest(x(t)=1/5*sin(2*t),de);

IiIh

g(t):=sin(3*t);

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGJyIiJA==

x_0:=1;v_0:=0;

IiIi

IiIh

de;ic;

LywmLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIjNiNJInhHRis2I0kidEdGKyIiIi1GMEYxIiIqLUkkc2luR0YoNiMsJEYyIiIk

NiQvLUkieEc2IjYjIiIhIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRig=

sol3:=dsolve({de,ic},x(t));

Ly1JInhHNiI2I0kidEdGJSwoLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEYnIiIkIyIiIiIjPS1JJGNvc0dGK0YuRjIqJkY0RjJGJ0YyIyEiIiIiJw==

sol3r:=rhs(sol3);

LCgtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIyIiIiIjPS1JJGNvc0dGJUYpRi4qJkYwRi5GK0YuIyEiIiIiJw==

plot(sol3r,t=0..13);

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plot(sol3r,t=0..20);

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g(t):=sin(2*t);

LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGJyIiIw==

sol4:=dsolve(de,x(t));

Ly1JInhHNiI2I0kidEdGJSwoKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRiciIiQiIiJJJF9DMkdGJUYyRjIqJi1JJGNvc0dGLEYvRjJJJF9DMUdGJUYyRjItRis2IywkRiciIiMjRjIiIiY=

sol4:=rhs(sol4);

LCgqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiQiIiJJJF9DMkdGKUYuRi4qJi1JJGNvc0dGJkYqRi5JJF9DMUdGKUYuRi4tRiU2IywkRiwiIiMjRi4iIiY=

xh:=select(has,sol4,3*t);

LCYqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiQiIiJJJF9DMkdGKUYuRi4qJi1JJGNvc0dGJkYqRi5JJF9DMUdGKUYuRi4=

xp:=select(has,sol4,2*t);

LCQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIjIyIiIiIiJg==

x_0:='x_0';v_0:='v_0';

SSR4XzBHNiI=

SSR2XzBHNiI=

eq1:=subs(t=0,xh+xp)=x_0;

LywoKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiISIiIkkkX0MyR0YqRi1GLSomLUkkY29zR0YnRitGLUkkX0MxR0YqRi1GLUYlI0YtIiImSSR4XzBHRio=

eq2:=subs(t=0,diff((xh+xp),t))=v_0;

LywoKiZJJF9DMkc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjIiIhRiciIiQqJkkkX0MxR0YmRictSSRzaW5HRipGLUYnISIkRigjIiIjIiImSSR2XzBHRiY=

eval(eq1);

L0kkX0MxRzYiSSR4XzBHRiQ=

eval(eq2);

LywmSSRfQzJHNiIiIiQjIiIjIiImIiIiSSR2XzBHRiU=

Exercise: Solve the equation: 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==. Identify the homogeneous solution and the particular solution. Is the solution periodic, if it is what is the period? Repeat the above for 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==.