Step 1:

According to the given information we get the equation as $y\large\frac{dy}{dx}$$ = x $

$ydy = xdx$

Integrating on both sides we get,

$\large\frac{y^2}{2} =\frac{ x^2}{2}$$ + C$

$y^2 - \large\frac{x^2}{2}$$ = C$

Step 2:

Since the curve passes through (0,-2), we substituting for $x$ and $y$ to find the value for C

$(-2)^2 - 0 = 2C$

$C =\large\frac{ 4}{2}$$ = 2$

Substituting this we get

$y^2 - x^2 = 4.$

This is the required solution.