Answer
$y^2 -2\ln{y} =x^2$ is a solution to the differential equation
Work Step by Step
Start by taking the derivative of the Solution
$y^2 -2\ln{y} =x^2$
$2y\frac{dy}{dx} - 2\frac{1}{y} \frac{dy}{dx}= 2x$
Then begin simplifying
$2(y-\frac{1}{y}) \frac{dy}{dx} =2x$ , divide both sides by two and the $(y- y^{-1}) $
$\frac{dy}{dx}= \frac{x}{y-\frac{1}{y}} $, then take a $\frac{1}{y}$ factor out of the denominator and put it in the numerator
$ \frac{dy}{dx}= \frac{xy}{y^2 -1}$
This satisfies the differential equation, so it is a solution
