Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 3 - Second Order Linear Equations - 3.4 Repeated Roots; Reduction of Order - Problems - Page 171: 7

Answer

$$ 4 y^{\prime \prime}+17 y^{\prime}+4 y=0 $$ The general solution of that equation is given by $$ y=c_{1} e^{-\frac{1}{4}t} +c_{2} e^{ -4t} $$ where $ c_{1} $ and $c_{2}$ are arbitrary constants.

Work Step by Step

$$ 4 y^{\prime \prime}+17 y^{\prime}+4 y=0 \quad \quad (1) $$ We assume that $ y = e^{rt}$, and it then follows that $r$ must be a root of the characteristic equation $$ 4r^{2}+17 r+4=0, $$ so its roots are $$ r_{1,2}=\frac{-17 \pm \sqrt{17^{2}-4 \cdot 4 \cdot 4}}{2 \cdot 4} $$ Thus the possible values of $r$ are $r_{1}=-\frac{1}{4} , r_{2}=-4$ ; the general solution of Eq. (1) is $$ y=c_{1} e^{-\frac{1}{4}t} +c_{2} e^{ -4t} $$ where $ c_{1} $ and $c_{2}$ are arbitrary constants.
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