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 513: 17

Answer

$y(x)=C_1e^{- x}\cos x+C_2 e^{-x}\sin x$

Work Step by Step

Solve the characteristic equation for the differential equation. $$r^2+2r+2=0$$ Factor and solve for the roots. $$r_1=-1-i ; r_2=-1+i$$ as roots. This implies that there are two independent solutions to the differential equation $y_1(x)=e^{- x}\cos x$ and $y_2= e^{-x}\sin x$. Therefore, the general equation is equal to $y(x)=C_1e^{- x}\cos x+C_2 e^{-x}\sin x$.
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