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# Assuming that the trajectory corresponding to a solution x = (t), y = (t), < t < , of an

ISBN: 9781119256007 392

## Solution for problem 24 Chapter 9.2

Elementary Differential Equations and Boundary Value Problems | 11th Edition

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Elementary Differential Equations and Boundary Value Problems | 11th Edition

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Problem 24

Assuming that the trajectory corresponding to a solution x = (t), y = (t), < t < , of an autonomous system is closed, show that the solution is periodic. Hint: Since the trajectory is closed, there exists at least one point ( x0, y0) such that (t0) = x0, (t0) = y0 and a number T > 0 such that (t0 + T ) = x0, (t0 + T ) = y0. Show that x = (t) = (t + T ) and y = (t) = (t + T ) is a solution, and then use the existence and uniqueness theorem to show that (t) = (t) and (t) = (t) for all t.

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