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: 21


$$y(x)=\frac{1}{x+1}\left(5+\ln x-\frac{1}{x}\right)$$

Work Step by Step

Given $$y^{\prime}+\frac{1}{x+1} y=x^{-2}, \quad y(1)=2$$ This a linear equation with $p(x) =\frac{1}{1+x}$ and $q(x) =x^{-2} $, so \begin{align*} \mu(x) &= e^{\int p(x)dx}\\ &=e^{\int\frac{1}{1+x} dx}\\ &= e^{\ln (1+x)}\\ &=1+x \end{align*} Hence the general solution is \begin{align*} \mu(x) y &=\int \mu (x)xq(x)dx\\ (1+x) y&=\int (1+x)x^{-2}dx \\ &=\int \frac{1}{x}+ x^{-2}dx\\ &=\ln x- x^{-1}+C \end{align*} Then $$ y(x)=\frac{1}{x+1}\left(C+\ln x-\frac{1}{x}\right)$$ Since $ y(1)= 2 $, then $C=5$, and hence $$y(x)=\frac{1}{x+1}\left(5+\ln x-\frac{1}{x}\right)$$
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.