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. Separation of Variables Separation of variables is an analytical method tha may be used to solve certain d.e.s (and certain i.v.p.s). At the heart of the method is the chain rule. Consider a (first order) differential equation that may be written as 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. That is, the derivative of the function is equal to the product of a function of only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and a function of only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== (the r.h.s. of the equation). If 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 we can write 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JJm1mcmFjR0YkNigtRiM2Iy1GLDYlUSNkeEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYsNiVRI2R0RidGOUY8LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEiPUYnL0Y9USdub3JtYWxGJy8lJmZlbmNlR0ZOLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRl9vLUYjNiUtRiw2JVEiZ0YnRjlGPC1GUDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZTRlVGV0ZZRmVuRmduRmluRltvL0Zeb1EmMC4wZW1GJy9GYW9GW3AtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGOUY8RlNGK0Yr. Recall LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRRjFGKw==(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==), so if we can find a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJIRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGNEY3Rj5GKw== with 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, using the chain rule we have that 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJIRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGNEY3Rj5GKw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw== = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GOjYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zmbi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUY6NiVRInhGJ0ZCRkVGNEY5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zmby8lKWJldmVsbGVkR0ZN LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSNkdEYnRjRGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRC8lKWJldmVsbGVkR1EmZmFsc2VGJw== so 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. Integrating both sides of the equation w.r.t. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtRiw2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEieEYnRjZGOUZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGXG8tRiM2JkYrLUYjNigtRj02LVEmJmludDtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYjNiUtRiw2JVEiZ0YnRjZGOUY8LUZXNiQtRiM2Iy1GLDYlUSJ0RidGNkY5RkBGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0ZmcC8lKmxpbmVicmVha0dRJWF1dG9GJy1GPTYtUTAmRGlmZmVyZW50aWFsRDtGJ0ZARkJGRUZHRklGS0ZNRk9GUUZURl5wLUY9Ni1RIitGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGZnEtRiw2JVEiY0YnRjZGOUYrRis= and since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtRiw2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEieEYnRjZGOUZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGXG8tRiM2KC1GPTYtUSYmaW50O0YnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtSSZtZnJhY0dGJDYoLUYjNiMtSSNtbkdGJDYkUSIxRidGQC1GIzYlRistRiM2JS1GLDYlUSJmRidGNkY5RjxGVkYrLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZocC8lKWJldmVsbGVkR0ZERistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGYnEvJSpsaW5lYnJlYWtHUSVhdXRvRictRj02LVEwJkRpZmZlcmVudGlhbEQ7RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZlbkYrRis=, the solution of the d.e. is given (implicitly) by 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. Note: the above is an implicit solution. Also note the arbitrary constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, this is the general solution. If we are also given an initial condition we can use it to solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and obtain a particular solution. Examples: 1. 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 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw==, seperating the variables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y1USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRy8lKWJldmVsbGVkR1EmZmFsc2VGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRI2R0RidGNEY3Ris= integrating 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 exponentiating 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 so 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 where 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 and absorbing the sign into the constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn we get the solution 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 another solution is 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. What is the interval of existence of these solutions? Exercise: verify that the above are solutions of the differential equation. 2. 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 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GKw== 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 a solution is given by 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 restart; int(1/x^2,x);

LCQqJEkieEc2IiEiIkYm

int(t,t);

LCQqJEkidEc2IiIiIyMiIiJGJg==

So a solution is 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. Solving for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== solve(-1/x = t^2/2+C,x);

LCQqJCwmKiRJInRHNiIiIiMiIiJJIkNHRidGKCEiIiEiIw==

we obtain 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 or 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. another solution is 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. What is the interval of existence of these solutions? What does it depend on? Exercise: verify that the above are solutions of the differential equation. 3. 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 rewrite the equation as 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 if 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 a solution is given by 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 int(1/(x^2+x),x);

LCYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YoIiIiLUYkNiMsJkYqRitGK0YrISIi

int(sin(t),t);

LCQtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKCEiIg==

So a solution is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNidGKy1GIzYlLUYsNiVRI2xuRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjsvJSZmZW5jZUdGOi8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZSLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEieEYnL0Y5USV0cnVlRicvRjxRJ2l0YWxpY0YnRjstRj82LVEoJm1pbnVzO0YnRjtGQkZERkZGSEZKRkxGTi9GUVEsMC4yMjIyMjIyZW1GJy9GVEZfby1GIzYlRjVGPi1GVjYkLUYjNiVGKy1GIzYlRlotRj82LVEiK0YnRjtGQkZERkZGSEZKRkxGTkZeb0Zgby1JI21uR0YkNiRRIjFGJ0Y7RitGO0YrLUY/Ni1RIj1GJ0Y7RkJGREZGRkhGSkZMRk4vRlFRLDAuMjc3Nzc3OGVtRicvRlRGZHAtRiM2Ji1GPzYtUSomdW1pbnVzMDtGJ0Y7RkJGREZGRkhGSkZMRk5GXm9GYG8tRiM2JS1GLDYlUSRjb3NGJ0Y4RjtGPi1GVjYkLUYjNiMtRiw2JVEidEYnRmduRmluRjtGaW8tRiw2JVEiQ0YnRmduRmluRitGKw== solve(ln(x)-ln(x+1) = -cos(t)+C,x);

LCQqJiwmISIiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJi1JJGNvc0dGKTYjSSJ0R0YsRiVJIkNHRixGJkYmRiVGJ0YmRiU=

so 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 solve(x^2+x=0);

NiQiIiEhIiI=

so additional solutions are 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 and 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. What is the interval of existence of these solutions? What does it depend on? Exercise: verify that the above are solutions of the differential equation. Problem: Solve the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JJm1mcmFjR0YkNigtRiM2Iy1GLDYlUSNkeEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYsNiVRI2R0RidGOUY8LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZJLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEiPUYnL0Y9USdub3JtYWxGJy8lJmZlbmNlR0ZOLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRl9vLUYyNigtRiM2Iy1GLDYlUSJ4RidGOUY8LUYjNiMtRiw2JVEidEYnRjlGPEZERkdGSkZMRis=.