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Ordinary Diff Eq Exam Review Solutions 37

Ordinary Diff Eq Exam Review Solutions 37 - 1 Solutions 12...

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Unformatted text preview: 1 Solutions 12. The right hand side is . Then 39 . Choose a rectangle about that lies above the -axis. Then both and are continuous on . Picard’s theorem applies and we can conclude there is a unique solution on an interval about . 13. The right hand side is . Then . Choose a rectangle about that contains no points on the line . Then both and are continuous on . Picard’s theorem applies and we can conclude there is a unique solution on an interval about . 14. The right hand side is condition , which is not deﬁned at the initial . Thus Picard’s theorem does not apply. 15. The corresponding integral equation is have We thus . . . We can write . We recognize this sum as the ﬁrst terms of the Taylor series expansion for . Thus the limiting function is . It is straightforward to verify that it is a solution. If then . Both and are continuous on the whole -plane. By Picard’s theorem, Theorem 5, is the only solution to the given initial value problem. 16. 1. The equation is separable so separate the variables to get . Integrating gives and the initial condition implies that the integration constant , so that the exact solution is 2. To apply Picard’s method, let and deﬁne ...
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