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.

restart:with(plots):

Warning, the name changecoords has been redefined

f1(t):=sin(2*t);

NiM+LUkjZjFHNiI2I0kidEc2Ii1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkidEc2IiIiIiIiIg==

f2(t):=cos(3*t);

NiM+LUkjZjJHNiI2I0kidEc2Ii1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJCIiIkkidEc2IiIiIiIiIg==

f3(t):=cos(5*t+2);

NiM+LUkjZjNHNiI2I0kidEc2Ii1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJiIiIkkidEc2IiIiIiIiIiIiIyIiIg==

f4(t):=sin(2^(1/2)*t);

NiM+LUkjZjRHNiI2I0kidEc2Ii1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiYpIiIjIyIiIiIiIyIiIkkidEc2IiIiIg==

f5(t):=cos(Pi*t);

NiM+LUkjZjVHNiI2I0kidEc2Ii1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJI1BpRyUqcHJvdGVjdGVkRyIiIkkidEc2IiIiIg==

f1(t);f2(t);f3(t);f4(t);f5(t);

NiMtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiMiIiJJInRHNiIiIiIiIiI= NiMtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiQiIiJJInRHNiIiIiIiIiI=

NiMtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiYiIiYiIiJJInRHNiIiIiIiIiIiIiMiIiI= NiMtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomKSIiIyMiIiIiIiMiIiJJInRHNiIiIiI= NiMtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSNQaUclKnByb3RlY3RlZEciIiJJInRHNiIiIiI=

plot(f1(t)+f2(t),t=0..8);

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

In order to better see the behavior of solutions of second order equations we can use the phase amplitude form of of periodic functions. That is given a periodic function of the form 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 we can write it as 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 where 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 and 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 (note this last equation has two solutions which differ by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=, you must chose the correct solution). Also recall the two trigonometric identities 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 and 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 using these identities we can often rewrite periodic functions to better see the phenomenon of beats. For example, consider the equation 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 with initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiP0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== restart:with(DEtools): sol:=rhs(dsolve({D((D)(x))(t)+9*x(t) = sin(2*t),x(0)=1,D(x)(0)=0},x(t)));

NiM+SSRzb2xHNiIsKComIyIiIyIjOiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJCIiIkkidEc2IiIiIiIiIiIiIiEiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJCIiIkkidEc2IiIiIiIiIiIiIiomIyIiIiIiJiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkidEc2IiIiIiIiIiIiIiIiIg==

Using the above identities we have that the solution can be written as 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 since sqrt((2/15)^2+1);evalf(arctan(2/15));

NiMsJComIyIiIiIjOiIiIikiJEgjIyIiIiIiIyIiIiIiIg== NiMkIitCYF5EOCEjNQ==

undampped oscillations Consider the differential equation 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. With initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtSSNtbkdGJDYkUSIwRidGQEZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGXW8tRmZuNiRRIjFGJ0ZARis= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=1. The characteristic equation is 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. This equation has the solutions: restart: solve(r^2+16=0);

NiQqJiIiJSIiIl4jRiVGJSwkRiMhIiI=

As you know, the homogeneous problem has the general solution 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. We now find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn c1c2:= {c1,c2};

NiM+SSVjMWMyRzYiPCRJI2MxRzYiSSNjMkc2Ig==

sol:=c1*cos(4*t)+c2*sin(4*t);

NiM+SSRzb2xHNiIsJiomSSNjMUc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiIiIiIiIiIiIiomSSNjMkc2IiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiIiIiIiIiIiIg==

eq1:=subs(t=0,sol);

NiM+SSRlcTFHNiIsJiomSSNjMUc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiIiIiKiZJI2MyRzYiIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiEiIiIiIiI=

eq2:=subs(t=0,diff(sol,t));

NiM+SSRlcTJHNiIsJiooIiIlIiIiSSNjMUc2IiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiISIiKigiIiUiIiJJI2MyRzYiIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiEiIiIiIiI=

solve({eq1=1,eq2=1},c1c2);

NiM8JC9JI2MyRzYiIyIiIiIiJS9JI2MxRzYiIiIi

sol:=cos(4*t)+(1/4)*sin(4*t);

NiM+SSRzb2xHNiIsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiIiIiIiIiomIyIiIiIiJSIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiIiIiIiIiIiIg==

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

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

underdampped oscillations Consider the differential equation 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. With initial conditions 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=1. The characteristic equation is 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. This equation has the solutions: restart: solve(r^2+2*r+16=0);

NiQsJiIiIiEiIiomXiNGJEYkKSIjOiNGJCIiI0YkRiQsJkYkRiUqJiwkRidGJEYkRihGJEYl

As you know, the homogeneous problem has the general solution 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. We now find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn c1c2:= {c1,c2};

NiM+SSVjMWMyRzYiPCRJI2MxRzYiSSNjMkc2Ig==

sol:=c1*exp(-t)*cos(sqrt(15)*t)+c2*exp(-t)*sin(sqrt(15)*t);

NiM+SSRzb2xHNiIsJiooSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHNiIhIiIiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomKSIjOiMiIiIiIiMiIiJJInRHNiIiIiIiIiIiIiIqKEkjYzJHNiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0RzYiISIiIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJikiIzojIiIiIiIjIiIiSSJ0RzYiIiIiIiIiIiIi

eq1:=subs(t=0,sol);

NiM+SSRlcTFHNiIsJiooSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiEiIiIiIiIqKEkjYzJHNiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiISIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiIiIi

eq2:=subs(t=0,diff(sol,t));

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

solve({eq1=1,eq2=1},c1c2);

NiM8JC9JI2MxRzYiIiIiL0kjYzJHNiIsJComIyIiIyIjOiIiIikiIzojIiIiIiIjIiIiIiIi

sol:=exp(-t)*(cos(4*t)+(2/sqrt(15))*sin(4*t));

NiM+SSRzb2xHNiIqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHNiIhIiIiIiIsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiIiIiIiIiooIyIiIyIjOiIiIikiIzojIiIiIiIjIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIiIlIiIiSSJ0RzYiIiIiIiIiIiIiIiIiIiIi

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

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

critically dampped oscillations Consider the differential equation 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. With initial conditions 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=1. The characteristic equation is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNihGKy1GIzYjLUklbXN1cEdGJDYlLUYsNiVRInJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRj9RJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIitGJ0ZFLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmluLUYjNiUtRkI2JFEiOEYnRkUtRks2LVExJkludmlzaWJsZVRpbWVzO0YnRkVGTkZRRlNGVUZXRllGZW4vRmhuUSYwLjBlbUYnL0Zbb0Zlb0Y4RkotRkI2JFEjMTZGJ0ZFLUZLNi1RIj1GJ0ZFRk5GUUZTRlVGV0ZZRmVuL0ZoblEsMC4yNzc3Nzc4ZW1GJy9GW29GXnAtRkI2JFEiMEYnRkVGKw==. This equation has the solutions: restart: solve(r^2+8*r+16=0);

NiQhIiVGIw==

As you know, the homogeneous problem has the general solution 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. We now find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn c1c2:= {c1,c2};

NiM+SSVjMWMyRzYiPCRJI2MyRzYiSSNjMUc2Ig==

sol:=c1*exp(-4*t)+c2*t*exp(-4*t);

NiM+SSRzb2xHNiIsJiomSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiEiIiIiIiIiIiooSSNjMkc2IiIiIkkidEc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiEiIiIiIiIiIg==

eq1:=subs(t=0,sol);

NiM+SSRlcTFHNiIqJkkjYzFHNiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiISIiIg==

eq2:=subs(t=0,diff(sol,t));

NiM+SSRlcTJHNiIsJiooIiIlIiIiSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiISIiKiZJI2MyRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiEiIiIiIiI=

solve({eq1=1,eq2=1},c1c2);

NiM8JC9JI2MxRzYiIiIiL0kjYzJHNiIiIiY=

sol:=exp(-4*t)+5*t*exp(-4*t);

NiM+SSRzb2xHNiIsJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJSIiIkkidEc2IiIiIiEiIiIiIiooIiImIiIiSSJ0RzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIiIlIiIiSSJ0RzYiIiIiISIiIiIiIiIi

plot(sol,t=0..2);

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

overdampped oscillations Consider the differential equation 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. With initial conditions 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=1. The characteristic equation is 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. This equation has the solutions: restart: solve(r^2+10*r+16=0);

NiQhIiMhIik=

As you know, the homogeneous problem has the general solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRiw2JVEiaEYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zmbi1GIzYnRistRiM2JS1GMjYlLUYsNiVRImNGJ0Y3RjotRiM2Iy1JI21uR0YkNiRRIjFGJ0ZJRkItRkY2LVExJkludmlzaWJsZVRpbWVzO0YnRklGS0ZORlBGUkZURlZGWC9GZW5RJjAuMGVtRicvRmhuRlxwLUklbXN1cEdGJDYlLUZGNi1RLyZFeHBvbmVudGlhbEU7RidGSUZLRk5GUEZSRlRGVkZYRltwL0ZoblEsMC4xMTExMTExZW1GJy1GIzYlLUZGNi1RKiZ1bWludXMwO0YnRklGS0ZORlBGUkZURlZGWC9GZW5RLDAuMjIyMjIyMmVtRicvRmhuRlxxLUYjNiUtRmVvNiRRIjJGJ0ZJRmhvLUYsNiVRInRGJ0Y3RjpGKy8lMXN1cGVyc2NyaXB0c2hpZnRHRkQtRkY2LVEiK0YnRklGS0ZORlBGUkZURlZGWEZbcUZdcS1GIzYlLUYyNiVGX28tRiM2I0ZgcUZCRmhvLUZfcDYlRmFwLUYjNiVGaHAtRiM2JS1GZW82JFEiOEYnRklGaG9GY3FGK0ZmcUYrRitGKw==. We now find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn c1c2:= {c1,c2};

NiM+SSVjMWMyRzYiPCRJI2MxRzYiSSNjMkc2Ig==

sol:=c1*exp(-2*t)+c2*exp(-8*t);

NiM+SSRzb2xHNiIsJiomSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkidEc2IiIiIiEiIiIiIiIiIiomSSNjMkc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiKSIiIkkidEc2IiIiIiEiIiIiIiIiIg==

eq1:=subs(t=0,sol);

NiM+SSRlcTFHNiIsJiomSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiIiIiKiZJI2MyRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiEiIiIiIiI=

eq2:=subs(t=0,diff(sol,t));

NiM+SSRlcTJHNiIsJiooIiIjIiIiSSNjMUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIhIiIiISIiKigiIikiIiJJI2MyRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiEiIiIhIiI=

solve({eq1=1,eq2=1},c1c2);

NiM8JC9JI2MyRzYiIyEiIiIiIy9JI2MxRzYiIyIiJCIiIw==

sol:=(3/2)*exp(-2*t)-(1/2)*exp(-8*t);

NiM+SSRzb2xHNiIsJiomIyIiJCIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkidEc2IiIiIiEiIiIiIiIiIiomIyIiIiIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiKSIiIkkidEc2IiIiIiEiIiIiIiEiIg==

plot(sol,t=0..2);

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