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 - 8.2 Constant Coefficient Homogeneous Linear Differential Equations - Problems - Page 514: 25

Answer

$y(x)=C_1e^{-x}+C_2xe^{-x}+C_3\cos \sqrt 3x+C_4 \sin \sqrt 3x$

Work Step by Step

Solve the auxiliary equation for the differential equation. $$(r^2+3)^2(r+1)=0$$ Factor and solve for the roots. $$(r+\sqrt 3i)(r-\sqrt 3i)(r+1)^2$$ Roots are: $r_1=-1$, as a multiplicity of $2$ and $r_2=-\sqrt 3i, r_3=\sqrt 3 i$ as a multiplicity of $1$. This implies that there are two independent solutions to the differential equation $y_1(x)=e^{-x}$ and $y_2=xe^{-x}$ and $y_3=\cos \sqrt 3x$ and $y_4= \sin \sqrt 3x$ Therefore, the general equation is equal to $y(x)=C_1e^{-x}+C_2xe^{-x}+C_3\cos \sqrt 3x+C_4 \sin \sqrt 3x$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.