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 - Page 21: 8

Answer

As given below

Work Step by Step

Take the derivatives of the function y. Remember to use chain rule. $y'(x)=-2c_1\sin2x+2c_2\cos2x$ $y''(x)=-4c_1\cos2x-4c_2\cos2x$ Substitute these equations into the differential equation to get: $y''+4y=0$ $-4c_1\cos2x-4c_2\cos2x +4(c_1\cos2x + c_2\sin2x)=0$ $-4c_1\cos2x-4c_2\cos2x+4c_1\cos2x + 4c_2\sin2x=0$ $0=0$ This satisfies the differential equation, so it must be a solution.
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