The given differential equation is:

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

We are tasked with finding the general solution of this equation.

Step 1: Rearrange the equation

First, rewrite the equation:

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

Now, divide both sides by $x^2 + y^2$:

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

Step 2: Separate variables

This equation is non-separable directly, but it's homogeneous, meaning the degree of both $x$ and $y$ is the same in the numerator and denominator.

Step 3: Use the substitution $v = \frac{y}{x}$

Let $y = vx$, so $dy = v \, dx + x \, dv$. Substitute this into the equation:

$\frac{dx}{dy} = \frac{2v}{1 + v^2}$

Now, solve this differential equation:

[ \frac{dx}{dy} = \frac{1}{2v} \implies f