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: 3

Answer

$y=c_{1}cos(\frac{1}{3}x)+c_{2}sin(\frac{1}{3}x)+\frac{1}{37}e^{2x}$

Work Step by Step

$9y''+y=e^{2x}$ $9r^{2}+1=0$ $r=i\sqrt{\frac{1}{9}}=\frac{1}{3}i$ $y=c_{1}cos(\frac{1}{3}x)+c_{2}sin(\frac{1}{3}x)$ $y_{p}=Ae^{2x}$ $y_{p}'=2Ae^{2x}$ $y_{p}''=4Ae^{2x}$ $9(4Ae^{2x})+Ae^{2x}=36Ae^{2x}+Ae^{2x}=37Ae^{2x}=e^{2x}$ $A=\frac{1}{37}$ $y=c_{1}cos(\frac{1}{3}x)+c_{2}sin(\frac{1}{3}x)+\frac{1}{37}e^{2x}$
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