Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems: 7

Answer

See below.

Work Step by Step

Take the derivatives of the function $y$. Remember to use chain rule. $$y'(x)=-5c_1e^{-5x}+5c_2e^{5x}$$ $$y''(x)=25c_1e^{-5x}+25c_2e^{5x}$$ Substitute these equations into the differential equation to get $$y''-25y=0$$ $$(25c_1e^{-5x}+25c_2e^{5x})-25(c_1e^{-5x}+c_2e^{5x})=0$$ This satisfies the differential equation, so it must be a solution.
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