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: 22

Answer

$y(x)=C_1e^{-4x}+C_2 e^{2x}+C_3e^{4x}$

Work Step by Step

Solve the auxiliary equation for the differential equation. $$(r-2)(r^2-16)=0$$ Factor and solve for the roots. $$(r-2)(r^2-16)^2=0$$ $$(r-2)(r-4)(r+4)=0$$ Roots are: $r_1=-4$ and $r_2=2$ and $r_3=4$ This implies that there are two independent solutions to the differential equation $y_1(x)=e^{-4x}$ and $y_2= e^{2x}$ and $y_3=e^{4x}$ Therefore, the general equation is equal to $y(x)=C_1e^{-4x}+C_2 e^{2x}+C_3e^{4x}$
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