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