Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 17 - Second-Order Differential Equations - 17.2 Nonhomogeneous Linear Equations - 17.2 Exercises - Page 1208: 19


a) $y=c_1\cos \dfrac{x}{2}+c_2\sin \dfrac{x}{2}-\dfrac{1}{3} \cos x$ b) $y=c_1\cos \dfrac{x}{2}+c_2\sin \dfrac{x}{2}-\dfrac{1}{3} \cos x$

Work Step by Step

a) $y_c=c_1\cos \dfrac{x}{2}+c_2\sin \dfrac{x}{2}\\ y_P=A \cos x+B \sin x$ Write the given differential equation becomes: $-3A \cos x-3B \sin x=\cos x$ This gives: $A=\dfrac{-1}{3}; B=0$ Then, we have $y=y_c+y_p$ or, $y=c_1\cos \dfrac{x}{2}+c_2\sin \dfrac{x}{2}-\dfrac{1}{3} \cos x$ b) $y_c=c_1\cos \dfrac{x}{2}+c_2\sin \dfrac{x}{2}$ This gives: $y_1=\cos \dfrac{x}{2}; y_2=\sin \dfrac{x}{2}$ $u_1=-\cos \dfrac{x}{2}-\dfrac{2}{3} \times (\cos \dfrac{x}{2})^3\\ u_2=\sin \dfrac{x}{2}-\dfrac{2}{3} \times (\sin \dfrac{x}{2})^3$ Now, $y_p=-\cos [\dfrac{x}{2}] -\dfrac{2}{3}(\cos \dfrac{x}{2})^3 \cos [\dfrac{x}{2}]+\sin \dfrac{x}{2}-\dfrac{2}{3}(\sin \dfrac{x}{2})^3\sin \dfrac{x}{2}=-\dfrac{1}{3} \times \cos x$ Then, we have $y=y_c+y_p$ or, $y=c_1\cos \dfrac{x}{2}+c_2\sin \dfrac{x}{2}-\dfrac{1}{3} \cos x$
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