Step 1: Rearrange the given equation.

The given differential equation is:

$$y \, dx + (x - y^2) \, dy = 0$$

Rearranging the terms, we get:

$$y \, dx = -(x - y^2) \, dy$$ $$y \, dx = (y^2 - x) \, dy$$

Dividing both sides by $dy$ (and assuming $y \neq 0$), we write it in terms of $\frac{dx}{dy}$:

$$y \frac{dx}{dy} = y^2 - x$$ $$y \frac{dx}{dy} + x = y^2$$

Step 2: Convert to standard linear form.

Divide the equation by $y$ to get the coefficient of $\frac{dx}{dy}$ as 1:

$$\frac{dx}{dy} + \frac{1}{y}x = y$$

This is a linear differential equation of the form $\frac{dx}{dy} + Px = Q$, where:

$$P = \frac{1}{y} \quad \text{and} \quad Q = y$$

Step 3: Find the Integrating Factor (I.F.).

The Integrating Factor is given by $e^{\int P \, dy}$.

$$I.F. = e^{\int \frac{1}{y} \, dy} = e^{\ln|y|} = y$$

Step 4: Write the general solution.

The solution of a linear differential equation is given by:

$$x(I.F.) = \int Q(I.F.) \, dy + C$$

Substituting the values:

$$x(y) = \int y(y) \, dy + C$$ $$xy = \int y^2 \, dy + C$$ $$xy = \frac{y^3}{3} + C$$

Step 5: Final Answer.

The general solution is:

$$xy = \frac{y^3}{3} + C$$

∴ $xy = \frac{y^3}{3} + C$