Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 8 - Linear Differential Equations of Order n - 8.10 Chapter Review - Additional Problems - Page 575: 12

Answer

$y(x)=C_1e^{x}\cos x+C_2xe^{x}\cos x+C_3x^2e^{x}\cos x+C_4e^{x}\sin x+C_5xe^{x}\sin x+C_6x^2e^x\sin x$

Work Step by Step

Given $(D^2-2D+2)^3y=0$ Solve the auxiliary equation for the differential equation. $(r^2-2r+2)^3=0$ Roots are: $r_1=1-i$ as a multiplicity of 3 and $r_2=1+i$ as a multiplicity of 3. This implies that there are two independent solutions to the differential equation $y_1(x)=e^{x}\cos x$ $y_2=xe^{x}\cos x$ and $y_3(x)=x^2e^{x}\cos x\\ y_4(x)=e^{x}\sin x\\ y_5(x)=xe^{x}\sin x \\ y_6(x)=x^2e^x\sin x$ Therefore, the general equation is equal to $y(x)=C_1e^{x}\cos x+C_2xe^{x}\cos x+C_3x^2e^{x}\cos x+C_4e^{x}\sin x+C_5xe^{x}\sin x+C_6x^2e^x\sin x$
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