Math 2650 - Linear Differential Equations 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 consider equations of the type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==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. For example take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiOUYnRj5GKw==, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJGRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GKw== (the roots of the characteristic polynomial are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRIklGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGKw== and 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) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtSSZtZnJhY0dGJDYoLUYjNiUtSSVtc3VwR0YkNiUtRiw2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GQlEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRkgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlMvJSlzdHJldGNoeUdGUy8lKnN5bW1ldHJpY0dGUy8lKGxhcmdlb3BHRlMvJS5tb3ZhYmxlbGltaXRzR0ZTLyUnYWNjZW50R0ZTLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXG8tRiw2JVEieEYnRj5GQS1GIzYlRistRiM2Iy1GOTYlLUYsNiVRI2R0RidGPkZBRkRGSkYrLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZgcC8lKWJldmVsbGVkR0ZTLUZONi1RIitGJ0ZIRlFGVEZWRlhGWkZmbkZobi9GW29RLDAuMjIyMjIyMmVtRicvRl5vRmlwLUYjNiUtRkU2JFEiOUYnRkhGTUZfb0YrLUZONi1RIj1GJ0ZIRlFGVEZWRlhGWkZmbkZobi9GW29RLDAuMjc3Nzc3OGVtRicvRl5vRmRxLUYjNiYtRkU2JFEiNEYnRkhGTS1GIzYlLUYsNiVRJGNvc0YnL0Y/RlNGSC1GTjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZIRlFGVEZWRlhGWkZmbkZobkZqbkZdby1JKG1mZW5jZWRHRiQ2JC1GIzYlRistRiM2JS1GLDYlUSgmb21lZ2E7RidGYHJGSEZNLUYsNiVRInRGJ0Y+RkFGK0ZIRitGK0Yr with initial conditions 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw== (0)=0. restart:with(DEtools): dsolve({D(D(x))(t)+9*x(t)=4*cos(omega*t),x(0)=0,D(x)(0)=0},x(t));

NiMvLUkieEc2IjYjSSJ0RzYiLCYqKCIiJSIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJCIiIkkidEc2IiIiIiIiIiIiIiwmIiIqISIiKiQpSSZvbWVnYUc2IiIiIyIiIiIiIiEiIiIiIiooIiIlIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkmb21lZ2FHNiIiIiJJInRHNiIiIiIiIiIsJiIiKiEiIiokKUkmb21lZ2FHNiIiIiMiIiIiIiIhIiIhIiI=

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.

sol(t):=-8/(-9+omega^2)*sin((3-omega)*t/2)*sin((3+omega)*t/2);

NiM+LUkkc29sRzYiNiNJInRHNiIsJCoqIiIpIiIiLCYiIiohIiIqJClJJm9tZWdhRzYiIiIjIiIiIiIiISIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJCooIyIiIiIiIyIiIiwmIiIkIiIiSSZvbWVnYUc2IiEiIiIiIkkidEc2IiIiIiIiIiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKCMiIiIiIiMiIiIsJiIiJCIiIkkmb21lZ2FHNiIiIiIiIiJJInRHNiIiIiIiIiIiIiIhIiI=

env(t):=-8/(-9+omega^2)*sin((3-omega)*t/2);

NiM+LUkkZW52RzYiNiNJInRHNiIsJCooIiIpIiIiLCYiIiohIiIqJClJJm9tZWdhRzYiIiIjIiIiIiIiISIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJCooIyIiIiIiIyIiIiwmIiIkIiIiSSZvbWVnYUc2IiEiIiIiIkkidEc2IiIiIiIiIiIiIiEiIg==

omega:=1:plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=2:plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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ONSTRAINEDG-%%ROOTG6'-%)BOUNDS_XG6#$"$+%!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%+I!""-%.BOUNDS_HEIGHTG6#$"%?D!""-%)CHILDRENG6"

omega:=5/2:plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=1.75*sqrt(2):plot([sol(t),env(t),-env(t)],t=0..20,color=[black,red,red]);

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omega:=29/10:plot([sol(t),env(t),-env(t)],t=0..50,color=[black,red,red]);

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For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjNGJ0Y3Ris= we get dsolve({D(D(x))(t)+9*x(t)=4*cos(3*t),x(0)=0,D(x)(0)=0},x(t));

NiMvLUkieEc2IjYjSSJ0RzYiLCQqKCMiIiMiIiQiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiQiIiJJInRHNiIiIiIiIiIiIiJJInRHNiIiIiIiIiI=

sol(t):=2/3*sin(3*t)*t;

NiM+LUkkc29sRzYiNiNJInRHNiIsJCooIyIiIyIiJCIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiJCIiIkkidEc2IiIiIiIiIiIiIkkidEc2IiIiIiIiIg==

env(t):=2/3*t;

NiM+LUkkZW52RzYiNiNJInRHNiIsJComIyIiIyIiJCIiIkkidEc2IiIiIiIiIg==

plot([sol(t),env(t),-env(t)],t=0..25,color=[black,red,red]);

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We consider equations of the type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==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 For example assume that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw==, 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, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJGRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj5GKw== (the roots of the characterisatic polynomial are 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 and 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). We now in addition assume that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRJDAuM0YnRjdGKw== and solve the initial value problem with initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==(0)=0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=0 restart:with(DEtools):with(plots):

Warning, the name changecoords has been redefined

de:=diff(x(t),t,t)+1/8*diff(x(t),t)+x(t)=3*cos(omega*t);

NiM+SSNkZUc2Ii8sKC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHNiItSSIkRyUqcHJvdGVjdGVkRzYkSSJ0RzYiIiIjIiIiKiYjIiIiIiIpIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInhHNiI2I0kidEc2IkkidEc2IiIiIiIiIi1JInhHNiI2I0kidEc2IiIiIiwkKiYiIiQiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSZvbWVnYUc2IiIiIkkidEc2IiIiIiIiIiIiIg==

dsolve({subs(omega=3/10,de),x(0)=0,D(x)(0)=0},x(t));

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

We now identify the steady state (the term that persists in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==) and the transient the term that decays in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. sol:=rhs(%);

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

transient:=select(has,sol,exp);

NiM+SSp0cmFuc2llbnRHNiIsJioqIyImISlbJCIoZGlEIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiMiIiIiIzsiIiJJInRHNiIiIiIhIiIiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKigjIiIiIiM7IiIiKSIkYiMjIiIiIiIjIiIiSSJ0RzYiIiIiIiIiIiIiKSIkYiMjIiIiIiIjIiIiISIiKigjIicrb1YiJ0BGOCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiMiIiIiIzsiIiJJInRHNiIiIiIhIiIiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKigjIiIiIiM7IiIiKSIkYiMjIiIiIiIjIiIiSSJ0RzYiIiIiIiIiIiIiISIi

steady:=remove(has,sol,exp);

NiM+SSdzdGVhZHlHNiIsJiomIyImKyE9IidARjgiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYjIiIkIiM1IiIiSSJ0RzYiIiIiIiIiIiIiIiIiKiYjIicrb1YiJ0BGOCIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiMiIiQiIzUiIiJJInRHNiIiIiIiIiIiIiIiIiI=

We now plot the solution, steady state and transient on the same coordinate system. plot({sol,transient,steady}, t=0..40,color=[magenta,red,blue]);

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and plot({sol,transient,steady}, t=0..80,color=[magenta,red,blue]);

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Note that the solution "converges" to the staedy state as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Yr. Also note that the steady state is periodic with period 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 which is the same as the period of the forcing function. Moreover, we observe that the steady state is not in phase with the forcing. To determine the phase difference between the steady state and forcing, we write the steady state in phase-amplitude form. s0:=simplify(subs(t=0,steady));

NiM+SSNzMEc2IiMiJytvViInQEY4

phase_amp:=A*cos(3/10*t-delta);

NiM+SSpwaGFzZV9hbXBHNiIqJkkiQUc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiMiIiQiIzUiIiJJInRHNiIiIiIhIiJJJmRlbHRhRzYiIiIiIiIi

p0:=simplify(subs(t=0, phase_amp));

NiM+SSNwMEc2IiomSSJBRzYiIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmRlbHRhRzYiIiIi

s1:=simplify(subs(t=0,diff(steady,t)));

NiM+SSNzMUc2IiMiJSthIidARjg=

p1:=simplify(subs(t=0, diff(phase_amp,t)));

NiM+SSNwMUc2IiwkKigjIiIkIiM1IiIiSSJBRzYiIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmRlbHRhRzYiIiIiIiIi

solve({p0=s0,p1=s1},{A,delta});

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

allvalues(%);

NiQ8JC9JJmRlbHRhRzYiLUknYXJjdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMjIiM6IiRrJC9JIkFHRiYsJComIyIlKzciJ0BGOCIiIilGNiNGNyIiI0Y3Rjc8JC9GMSwkRjMhIiIvRiUsJkYnRjdJI1BpR0YqRj4=

evalf(%);

NiQ8JC9JIkFHNiIkIislbzJSSCQhIiovSSZkZWx0YUdGJiQiK14pWyY9VCEjNjwkL0YrJCErbHJTK0pGKS9GJSQhKyVvMlJIJEYp

The steady state is displaced in phase by about 0.0412 from the forcing. Next we show that the amplitude and phase of the steady state are independent of the initial conditions (hence the initial conditions affect only the transient). To do this we find a general solution of the equation and show that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn appear only in the transient terms. dsolve({subs(omega=3/10,de),x(0)=x_0,D(x)(0)=x_1},x(t));

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

combine(%,trig): collect(%,exp);

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

We now consider the general equation where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= is a fixed positive constant. We solve the equation with that initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw==(0)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn. We again identify the steady state and show it is independent of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn. de;

NiMvLCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieEc2IjYjSSJ0RzYiLUkiJEclKnByb3RlY3RlZEc2JEkidEc2IiIiIyIiIiomIyIiIiIiKSIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHNiJJInRHNiIiIiIiIiItSSJ4RzYiNiNJInRHNiIiIiIsJComIiIkIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkmb21lZ2FHNiIiIiJJInRHNiIiIiIiIiIiIiI=

dsolve({de,x(0)=x_0,D(x)(0)=x_1},x(t));

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

combine(%,trig): gen_sol:=collect(rhs(%),exp);

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

steady:=simplify(remove(has,%,exp));

NiM+SSdzdGVhZHlHNiIsJCooIiNDIiIiLCgqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJJm9tZWdhRzYiIiIiSSJ0RzYiIiIiIiIiSSZvbWVnYUc2IiIiIiEiIiomIiIpIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkmb21lZ2FHNiIiIiJJInRHNiIiIiIiIiIhIiIqKCIiKSIiIilJJm9tZWdhRzYiIiIjIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkmb21lZ2FHNiIiIiJJInRHNiIiIiIiIiIiIiIiIiIsKCIjayIiIiomIiRGIiIiIilJJm9tZWdhRzYiIiIjIiIiISIiKiYiI2siIiIpSSZvbWVnYUc2IiIiJSIiIiIiIiEiIiEiIg==

We now write the steady state in the phase amplitude form. s0:=simplify(subs(t=0,steady));

NiM+SSNzMEc2IiwkKigiJCM+IiIiLCYiIiIhIiIqJClJJm9tZWdhRzYiIiIjIiIiIiIiIiIiLCgiI2siIiIqJiIkRiIiIiIpSSZvbWVnYUc2IiIiIyIiIiEiIiomIiNrIiIiKUkmb21lZ2FHNiIiIiUiIiIiIiIhIiIhIiI=

phase_amp:=A*cos(omega*t-delta);

NiM+SSpwaGFzZV9hbXBHNiIqJkkiQUc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJkkmb21lZ2FHNiIiIiJJInRHNiIiIiIiIiJJJmRlbHRhRzYiISIiIiIi

p0:=simplify(subs(t=0,phase_amp));

NiM+SSNwMEc2IiomSSJBRzYiIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmRlbHRhRzYiIiIi

s1:=simplify(subs(t=0,diff(steady,t)));

NiM+SSNzMUc2IiwkKigiI0MiIiIpSSZvbWVnYUc2IiIiIyIiIiwoIiNrIiIiKiYiJEYiIiIiKUkmb21lZ2FHNiIiIiMiIiIhIiIqJiIjayIiIilJJm9tZWdhRzYiIiIlIiIiIiIiISIiIiIi

p1:=simplify(subs(t=0,diff(phase_amp,t)));

NiM+SSNwMUc2IiooSSJBRzYiIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmRlbHRhRzYiIiIiSSZvbWVnYUc2IiIiIg==

solve({p0=s0,p1=s1},{A,delta});

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

allvalues(%);

NiQ8JC9JIkFHNiIsJComIiNDIiIiKSokLCgiI2tGKiomIiRGIkYqKUkmb21lZ2FHRiYiIiNGKiEiIiomRi5GKilGMiIiJUYqRipGNCNGKkYzRipGKi9JJmRlbHRhR0YmLUknYXJjdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiQqJkYyRipGK0YqKiYsJiIiKUYqKiZGREYqRjFGKkY0RipGK0YqPCQvRjotRjw2JCwkRkFGNCwkRkJGNC9GJSwkRihGNA==

assign(%[1]); We take a value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= corresponding to a positive value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. We now compute the phase lag for different LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=. evalf(subs(omega=1,delta));

NiMkIitGanpxOiEiKg==

evalf(subs(omega=2,delta));

NiMkIitBOVhlSSEiKg==

evalf(subs(omega=3,delta));

NiMkIitUPnYlNCQhIio=

So the phase lag is approximately 1.571 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjFGJ0Y3Ris=, 3.058 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjJGJ0Y3Ris=, and 3.095 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjNGJ0Y3Ris=. plot(delta,omega=0.1..10);

NigtJSdDVVJWRVNHNiQ3YG83JCQiIiIhIiIkIjElW0c/PGZERSIhIzw3JCQiLERFPXo6JCEjNiQiMUYzZ2YnWz1RJSEjPDckJCIyLl1QTSRbcic0JSEjPCQiMlZoJz5EIz1mOSchIz03JCQiLnZWUzZiLiYhIzgkIjEkel0iel1vNyUpISM8NyQkIi9EMU1qRyI0JyEjOSQiMm9mKmVxLXMvNyEjPDckJCItdmo3MVpyISM3JCIyQ0NuXjdDaiE9ISM8NyQkIi52JDR6V3l3ISM4JCIyRit4VXNRdEgjISM8NyQkIjEsKytiWCQpNCMpISM7JCIyLmNMJCozZSlcSSEjPDckJCIxLF1pKzdBVCgpISM7JCIyRCpbVEgvWlBWISM8NyQkIjEtK0RZeWdzIyohIzskIjFWP3FteHMzcCEjOzckJCIxbFNtNUojW1MqISM7JCIxTVgneSlHb1V6ISM7NyQkIjFFInldUFFxYCohIzskIjBaODpPI1I8IyohIzo3JCQiMSo9I1xST0RwJyohIzskIjJjcVJYJkcteDUhIzs3JCQiMV9pIVIhKm85ISkqISM7JCIyJTRyOjB0SWc3ISM7NyQkIjE5LktvVG9MKiohIzskIjIiUUEjPklPWlkiISM7NyQkIjJ2VnRLJSoqZTE1ISM7JCIyMjh3cSFmXXY7ISM7NyQkIjJQJVtycDkiKT41ISM7JCIxWCQ9O0dnXCg9ISM6NyQkIjBEY2gqSC5MNSEjOSQiMnhXMGt0dy4wIyEjOzckJCIyRjFSIVxnWmY1ISM7JCIyWDkybWctc0ojISM7NyQkIjF2PSM+NT5mMyIhIzokIjAiW1gqKXlQJFwjISM5NyQkIjJ3by9bOmlCNiIhIzskIjJ0VnZdRmg7aCMhIzs3JCQiMi1dKG8yXyEpUTYhIzskIjBzS3lrJVslcCMhIzk3JCQiL0RjKGVwb0IiISM4JCIyRC9QT1x3dyZHISM7NyQkIi92Vm5SJFxMIiEjOCQiMiVcUSNbdnQ4JEghIzs3JCQiL3ZvX1JaTzkhIzgkIjJ5TCVmc2FLdUghIzs3JCQiMi1dUHokUixROiEjOyQiMVZwPC4mPTwrJCEjOjckJCIyLF1QZnRFSWsiISM7JCIxRm0qZTJLOC0kISM6NyQkIjIuXVBSYFIhWzwhIzskIjIncHN4c11wTkkhIzs3JCQiMi1dKD12OlJkPiEjOyQiMTE7L208UmJJISM6NyQkIjIwK11BKWV0c0AhIzskIjF0USh5Z0cob0khIzo3JCQiMi8rRF5FNkNPIyEjOyQiMihvN0VBJD5zMiQhIzs3JCQiMjErXTwjMyVmZCMhIzskIjI4IlEwVHpeJTMkISM7NyQkIi1EXnJNIXojISM2JCIyOjxwJWUiUi00JCEjOzckJCIyLytdI3l0J3AqSCEjOyQiMUZiLmZGcCU0JCEjOjckJCIvdiQqUSQpZiU9JCEjOCQiMVZLaFddMik0JCEjOjckJCIyLStdIypwNHhTJCEjOyQiMih6Jm8qM2laLEohIzs3JCQiMjErK3FHOW5mJCEjOyQiMW5GdC5cJVI1JCEjOjckJCIyMF1QV2JybCJRISM7JCIxM0d6MDVXMUohIzo3JCQiLCYqWyM9NlMhIzUkIjF3LSlHIW9QM0ohIzo3JCQiMjBdUCVIOnFDVSEjOyQiMjleXXcnKWUtNiQhIzs3JCQiL0RKW0AtR1chIzgkIjIkKTQyQCxiPTYkISM7NyQkIi52VillO1NZISM3JCIxJT1OVFFcTDYkISM6NyQkIi92PWUrKVwkWyEjOCQiMVgiPSM0MWY5SiEjOjckJCIuREpZOF4vJiEjNyQiMidRUycpKXAzZTYkISM7NyQkIi92JFIsJFFqXyEjOCQiMjIjPnAlR2ZwNiQhIzs3JCQiL0RjKkgoUWBhISM4JCIyPWVURit5eTYkISM7NyQkIi52TDgoZmVjISM3JCIxYiZwbz0lej1KISM6NyQkIiw6KCopZnFlISM1JCIyWWEmKnlMbic+SiEjOzckJCIudik0OSt5ZyEjNyQiMkwlZidIamQvNyQhIzs3JCQiL3ZWTFJueWkhIzgkIjFPJ3osbm82NyQhIzo3JCQiLkRyYidbLGwhIzckIjImPkhzXUghPjckISM7NyQkIjImKioqKlw0T3AsbiEjOyQiMSdlRVBAPUQ3JCEjOjckJCIyMCtdN1BkYSJwISM7JCIxXlsjeUVMSjckISM6NyQkIi92JDQ2ZiI0ciEjOCQiMi94SUJxY083JCEjOzckJCIxLCt2dmkjNEsoISM6JCIyMDZUWVYmPkNKISM7NyQkIi92VnVFPT92ISM4JCIyVnY8UTd0WTckISM7NyQkIjIwXSg9Q0lZR3ghIzskIjJPT2MjZVo5REohIzs3JCQiLURPXjRLeiEjNiQiMmBrJVJdMmVESiEjOzckJCIvdiRmM3pfOSkhIzgkIjF0VC5gRixFSiEjOjckJCIrbjBnXSQpISIqJCIxYTZzXnFTRUohIzo3JCQiLkRtVnIwYykhIzckIjEoUjp5KikqeUVKISM6NyQkIi92VlRPISlvKCkhIzgkIjImMytYVjI6RkohIzs3JCQiMSwrXTZ1OWcqKSEjOiQiMCN5RXNwWUZKISM5NyQkIjEsXTdsKlsleiIqISM6JCIxXyg9R2c3eTckISM6NyQkIisqKVtmdiQqISIqJCIyMGZcTmMyIkdKISM7NyQkIi92VmF0cyVlKiEjOCQiMU5VYDckMyVHSiEjOjckJCIvRGMzUSpbeSohIzgkIjJqejhCdSRvR0ohIzs3JCQiJCsiISIiJCIyJjMmcD0xbipHSiEjOy0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUlVklFV0c2JDskIiIiISIiJCIkKyIhIiI7JCExaSpwI29QN3FcISM8JCIyRVxzNypRSCI+JCEjOy0lK0FYRVNMQUJFTFNHNidRJm9tZWdhNiJRITYiLSUlRk9OVEc2JSUhRyUhRyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUqQVhFU1NUWUxFRzYjJSdOT1JNQUxHLSUoU0NBTElOR0c2IyUuVU5DT05TVFJBSU5FREctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiQrIyEiIi0lKUJPVU5EU19ZRzYjJCIjXSEiIi0lLUJPVU5EU19XSURUSEc2IyQiJUlEISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJT88ISIiLSUpQ0hJTERSRU5HNiI=

We also determine the amplitude of the steady state as a function of the forcing frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=. A;

NiMsJComIiNDIiIiKSokLCgiI2siIiIqJiIkRiIiIiIpSSZvbWVnYUc2IiIiIyIiIiEiIiomIiNrIiIiKUkmb21lZ2FHNiIiIiUiIiIiIiIhIiIjIiIiIiIjIiIiIiIi

plot(A,omega=0.1..10);

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

We finally find the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= for which the amplitude is maximal. Aprime:=diff(A,omega);

NiM+SSdBcHJpbWVHNiIsJCoqIiM3IiIiKSokLCgiI2siIiIqJiIkRiIiIiIpSSZvbWVnYUc2IiIiIyIiIiEiIiomIiNrIiIiKUkmb21lZ2FHNiIiIiUiIiIiIiIhIiIjIiIiIiIjISIiKSwoIiNrIiIiKiYiJEYiIiIiKUkmb21lZ2FHNiIiIiMiIiIhIiIqJiIjayIiIilJJm9tZWdhRzYiIiIlIiIiIiIiIiIjISIiLCYqJiIkYSMiIiJJJm9tZWdhRzYiIiIiISIiKiYiJGMjIiIiKUkmb21lZ2FHNiIiIiQiIiIiIiIiIiIhIiI=

solve(Aprime=0,omega);

NiUiIiEsJComIyIiIiIjO0YnKSIkYSMjRiciIiNGJ0YnLCRGJSEiIg==

evalf(%);

NiUkIiIhRiQkIisxNCczJyoqISM1JCErMTQnMycqKkYn

The maximum amplitude occurs when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= is approximately 0.9961. This value is close to 1 at which the corresponding undamped oscillator exhibits resonance. Also not that we exclude LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjBGJ0Y3Ris= since then the forcing is constant. subs(omega=.9960860906,A);

NiMkIistOHEvQyEiKQ==

Also note that unlike resonance the amplitude for this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= is still finite (approximately 24.05).