Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 10 - Introduction to Differential Equations - 10.4 First-Order Linear Equations - Exercises - Page 524: 24


$$ y(x)=1 $$

Work Step by Step

Given$$y^{\prime}+(\sec t) y=\sec t, \quad y\left(\frac{\pi}{4}\right)=1$$ This is a linear equation with $p(t) =\sec t\ \ q(t) =\sec t$, so \begin{align*} \mu(t)&=e^{\int p(t)dt}\\ &=e^{\int \sec t dt}\\ &=e^{\ln |\sec t+\tan t |}\\ &=\sec t+\tan t \end{align*} Then \begin{align*} y\mu(t) &=\int \mu(t)q(t)dt\\ (\sec t+\tan t ) y &=\int (\sec^2 t+\tan t\sec t ) dt\\ &= \sec t+\tan t+C \end{align*} Then $$y=1+\frac{C}{\sec t+\tan t}$$ Since $y(\pi/4)=1 $, then $C=0$, and hence $$ y(x)=1 $$
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.