Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 9 - Differential Equations - 9.3 Separable Equations - 9.3 Exercises - Page 645: 3


$\displaystyle y=±\sqrt{x^2+2\ln|x|+C}$

Work Step by Step

$\displaystyle xyy'=xy\cdot\frac{dy}{dx}=x^2+1$ $\displaystyle \frac{dy}{dx}=\frac{x^2+1}{xy}$ Seperation of variables: $\displaystyle ydy=(\frac{x^2+1}{x})dx=(\frac{x^2}{x}+\frac{1}{x})dx=(x+\frac{1}{x})dx$ Integrate both sides $\displaystyle \int ydy=\int x+\frac{1}{x}dx$ $\displaystyle \frac{y^2}{2}=\frac{x^2}{2}+\ln|x|+C$ $\displaystyle y^2=x^2+2\ln|x|+C$ $\displaystyle y=±\sqrt{x^2+2\ln|x|+C}$ The reason why the answer has $C$ and not $2C$ when $2$ is multiplied on both sides is because $C$, the constant of integration, has not taken on a definite value. Also, the answer MUST have the plus-minus symbol because the solution could either be positive or negative depending on the initial value.
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