Math 2650 - Linear Differential Equations

A. J. Meir

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.

Examples and Things to Come

An ordinary differential equation is an equation which involves an unknown function and its derivatives with respect to one independent variable. (A partial differential equation involves an unknown function and its partial derivatives with respect to several independent variables.)

The pendulum

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZBLUYjNictRjQ2I1EhRictRiM2JS1GMTYlLUY0NiVRI2R0RidGN0Y6Rj1GQ0ZGRkFGS0ZGRkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlovJSliZXZlbGxlZEdRJmZhbHNlRidGRkZB LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEibEYnRjRGN0Y6Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkkvJSliZXZlbGxlZEdRJmZhbHNlRidGOkY9 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEjZHRGJ0Y0RjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy + 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 = F(t)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZBLUYjNictRjQ2I1EhRictRiM2JS1GMTYlLUY0NiVRI2R0RidGN0Y6Rj1GQ0ZGRkFGS0ZGRkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlovJSliZXZlbGxlZEdRJmZhbHNlRidGRkZB LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEibEYnRjRGN0Y6Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkkvJSliZXZlbGxlZEdRJmZhbHNlRidGOkY9 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEjZHRGJ0Y0RjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy + 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy = F(t)

Population growth

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEjZHRGJ0Y0RjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPQ== p = birth rate - death rate

= b p -d p

Radioactive decay

In general an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYnth order o.d.e. is an equation of the form

F(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R0RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0=, 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,...,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) = 0

and a general first order o.d.e. is an equation of the form

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnL0Y4USdub3JtYWxGJy1GIzYlLUYxNiVRI2R0RidGNEY3RjpGPS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Y6Rj0= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJ0RidGNEY3LUY7Ni1RIixGJ0Y+RkAvRkRGNkZFRkdGSUZLRk1GTy9GU1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ4RidGNEY3LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+Rj5GX29GPkYrRl9vRj4=

A solution of a differential equation is a differentiable function defined on some interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn which satisfies the equation (the o.d.e.) at every point in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn. The solution may given explicitly or implicitly. That is this function may be defined explicitly or implicitly.

An explicit solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy is a solution of the form

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 = expression involving only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn (the independent variable)

and an implicit solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy is of the form

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictRiw2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGOS8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlAtSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJ0RicvRjdRJXRydWVGJy9GOlEnaXRhbGljRictRj02LVEiLEYnRjlGQC9GQ0ZmbkZERkZGSEZKRkxGTi9GUlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ4RidGZW5GZ24vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjlGOUZib0Y5LUY9Ni1RIj1GJ0Y5RkBGQkZERkZGSEZKRkwvRk9RLDAuMjc3Nzc3OGVtRicvRlJGaW8tSSNtbkdGJDYkUSIwRidGOUZib0Y5RitGYm9GOQ==

(an expression involving both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, the dependent and independent variables) it implicitly defines the function 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 (the solution of the differential equation).

restart:with(plottools):with(plots):

Consider the radioactive decay problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEjZHRGJ0Y0RjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPQ== q = -k q

eq1:=diff(q(t),t)=-k*q(t);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJxRzYiNiNJInRHRilGKywkKiZJImtHRikiIiJGJ0YvISIi

dsolve(eq1,q(t));

Ly1JInFHNiI2I0kidEdGJSomSSRfQzFHRiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZJImtHRiVGKkYnRiohIiJGKg==

ic1:=q(0)=q[0];

Ly1JInFHNiI2IyIiISZGJEYm

dsolve({eq1,ic1},q(t));

Ly1JInFHNiI2I0kidEdGJSomJkYkNiMiIiEiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZJImtHRiVGLEYnRiwhIiJGLA==

soln1:=rhs(%);

KiYmSSJxRzYiNiMiIiEiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZJImtHRiVGKEkidEdGJUYoISIiRig=

q[0]:=2;k:=0.1;

IiIj

JCIiIiEiIg==

soln1;

LCQqJiIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiRGJSEiIkYlSSJ0R0YrRiVGMEYlRiU=

plot(soln1,t=0..100,thickness=2);

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

Consider the population growth LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEjZHRGJ0Y0RjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPQ== p = b p - d p

eq2:=diff(p(t),t)=b*p(t)-d*p(t);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJwRzYiNiNJInRHRilGKywmKiZJImJHRikiIiJGJ0YvRi8qJkkiZEdGKUYvRidGLyEiIg==

dsolve(eq2,p(t));

Ly1JInBHNiI2I0kidEdGJSomSSRfQzFHRiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomLCZJImJHRiVGKkkiZEdGJSEiIkYqRidGKkYq

ic2:=p(0)=p[0];

Ly1JInBHNiI2IyIiISZGJEYm

dsolve({eq2,ic2},p(t));

Ly1JInBHNiI2I0kidEdGJSomJkYkNiMiIiEiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomLCZJImJHRiVGLEkiZEdGJSEiIkYsRidGLEYs

soln2:=rhs(%);

KiYmSSJwRzYiNiMiIiEiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomLCZJImJHRiVGKEkiZEdGJSEiIkYoSSJ0R0YlRihGKA==

p[0]:=2;b:=3;d:=2;

IiIj

IiIk

IiIj

soln2;

LCQqJiIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0YrRiVGJQ==

plot(soln2,t=0..1,thickness=2);

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

Consider the pendulum LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0ZBLUYjNictRjQ2I1EhRictRiM2JS1GMTYlLUY0NiVRI2R0RidGN0Y6Rj1GQ0ZGRkFGS0ZGRkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlovJSliZXZlbGxlZEdRJmZhbHNlRidGRkZB LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEibEYnRjRGN0Y6Rj0vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkkvJSliZXZlbGxlZEdRJmZhbHNlRidGOkY9 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjhRJ25vcm1hbEYnLUYjNiUtRjE2JVEjZHRGJ0Y0RjdGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYnLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUS1GMTYlUSJnRidGNEY3LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+LUYjNiUtRjE2JVEibEYnRjRGN0ZXRj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5vLyUpYmV2ZWxsZWRHRkJGV0Y+ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy = 0

m:='m';c:='c';l:='l';g:=10;

SSJtRzYi

SSJjRzYi

SSJsRzYi

IiM1

eq3:=m*diff(phi(t),t,t)+(c/l)*diff(phi(t),t)+(m*g/l)*phi(t)=0;

LywoKiZJIm1HNiIiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYpNiQtSSRwaGlHRiY2I0kidEdGJkYxRjFGJ0YnKihJImNHRiZGJ0kibEdGJiEiIkYsRidGJyoqIiM1RidGJUYnRjRGNUYuRidGJyIiIQ==

dsolve(eq3,phi(t));

Ly1JJHBoaUc2IjYjSSJ0R0YlLCYqJkkkX0MxR0YlIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJCosI0YrIiIjRissJkkiY0dGJUYrKiQpLCYqJClGN0Y1RitGKyooIiNTRispSSJtR0YlRjVGK0kibEdGJUYrISIiRjRGK0ZCRitGQUZCRkBGQkYnRitGQkYrRisqJkkkX0MyR0YlRistRi02IywkKixGNEYrLCZGN0YrRjhGK0YrRkFGQkZARkJGJ0YrRkJGK0Yr

m:=1/10;c:=0;l:=4;g:=10;

IyIiIiIjNQ==

IiIh

IiIl

IiM1

eq3:=m*diff(phi(t),t,t)+(c/l)*diff(phi(t),t)+(m*g/l)*phi(t)=0;

LywmKiYjIiIiIiM1RiYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYpNiQtSSRwaGlHNiI2I0kidEdGMEYyRjJGJkYmKiYjRiYiIiVGJkYuRiZGJiIiIQ==

ic3:=phi(0)=1,D(phi)(0)=0;

NiQvLUkkcGhpRzYiNiMiIiEiIiIvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRidGKA==

dsolve({eq3,ic3},phi(t));

Ly1JJHBoaUc2IjYjSSJ0R0YlLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJCooIyIiIiIiI0YxKSIjNUYwRjFGJ0YxRjE=

soln3:=rhs(%);

LUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJCooIyIiIiIiI0YsKSIjNUYrRixJInRHRidGLEYs

plot(soln3,t=0..15,thickness=2);

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

p:=unapply(soln3,t);

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJCooIyIiIiIiI0YyKSIjNUYxRjJGJEYyRjJGJUYlRiU=

L:=(x,y)->line([0,0],[x,y],thickness=3,color=red):

N:=50;

IiNd

x:=l*sin(p((15/N)*n));y:=-l*cos(p((15/N)*n));

LCQqJiIiJSIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkkY29zR0YoNiMsJCooIyIiJCIjP0YlKSIjNSNGJSIiI0YlSSJuR0YrRiVGJUYlRiU=

LCQqJiIiJSIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUYnNiMsJCooIyIiJCIjP0YlKSIjNSNGJSIiI0YlSSJuR0YrRiVGJUYlISIi

seq1:=seq(L(x,y),n=0..N):

display(seq1,insequence=true);

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