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 1207: 16

Answer

$y_p(x)=x(A x^3+Bx^+Cx+D)e^{x}$

Work Step by Step

Consider $G(x)=e^{\alpha x} A(x) \sin mx $ or $G(x)=e^{\beta x} A(x) \cos mx $ The trial solution for the method of undetermined coefficients is defined as: $y_p(x)=e^{\alpha x} B(x) \sin mx +e^{\beta x} C(x) \cos mx$ When the sum of the coefficients of a differential equation is zero. Then, we have $y_p(x)=x e^{\alpha x} B(x)$ On comparing the above equation with the given equation, we get the sum of the coefficients of a differential equation is $1+3-4=0$. Thus, the trial solution for the method of undetermined coefficients is: $y_p(x)=x(A x^3+Bx^+Cx+D)e^{x}$
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