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 2 - First Order Differential Equations - 2.1 Linear Equations; Method of Integrating Factors - Problems - Page 40: 27

Answer

$t=1.364,\ y=0.820$

Work Step by Step

Finding the integrating factor: $\mu(t)=\exp(\int 1/2\ dt)$ $\mu(t)=e^{t/2}$ Finding the y-function: $y=\frac{1}{\mu(t)}[\int \mu(s)(2cos(s))ds+c]$ $y=e^{-t/2}[4/5\ e^{t/2}(2\sin t+\cos t)+c]$ $y=4/5(2\sin t+\cos t)+ce^{-t/2}$ $y(0)=4/5+c=-1$ $c=-9/5$ Local maximum, $y'=0\rightarrow 4/5(2\cos t-\sin t)+9/10\ e^{-t/2}=0$ The first solutions is $t=1.364,\ y=0.820$
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