Answer
False
Work Step by Step
Given $D(D^2-4)^2y=0$
and $y(x)=c_1e^{-3x}+c_2xe^{-3x}+c_3\cos 5x+c_4\sin 5x$
Obtain: $D(D^2-4)^2y\\
=(D^5-8D^3+16D)y\\
=D^5y-8D^3y+16Dy$
Substituting: $0-0+16=16$
Hence, $y(x)$ is not a solution for the equation.
Differential Equations and Linear Algebra (4th Edition)
by Goode, Stephen W.; Annin, Scott A.
Published by
Pearson
ISBN 10:
0-32196-467-5
ISBN 13:
978-0-32196-467-0
Chapter 8 - Linear Differential Equations of Order n - 8.2 Constant Coefficient Homogeneous Linear Differential Equations - True-False Review - Page 513: e
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