Clever!

And simpler than mine, no less.

However I'm going to make two changes to the above. First your solution is $\displaystyle y(x) = Cx^2 + C^2$ and I am going to use the derivative equation to eliminate C, not x.

$\displaystyle y = Cx^2 + C^2$

$\displaystyle y^{\prime} = 2Cx \implies C = \frac{y^{\prime}}{2x}$

Thus

$\displaystyle y = \left ( \frac{y^{\prime}}{2x} \right ) x^2 + \left ( \frac{y^{\prime}}{2x} \right ) ^2$

$\displaystyle y = \frac{1}{2}x y^{\prime} + \frac{1}{4x^2}(y^{\prime}) ^2$

I would then multiply both sides by $\displaystyle x^2$ since the solution doesn't have the restriction on x = 0.

$\displaystyle 4x^2y = 2x^3 y^{\prime} + (y^{\prime}) ^2$

(I threw in an extra factor of 4 to clear the fraction and clean it up a little.)

-Dan