Solve the following differential equation:

$x^2 dy + (xy + y^2) dx = 0 \implies \frac{dy}{dx} = -\frac{xy + y^2}{x^2}$. Let $y = vx$, then $\frac{dy}{dx} = v + x \frac{dv}{dx}$. $v + x \frac{dv}{dx} = -(v + v^2) \implies x \frac{dv}{dx} = -2v - v^2 = -v(v + 2)$. Separating variables: $\frac{dv}{v(v + 2)} = -\frac{dx}{x} \implies \frac{1}{2}\left(\frac{1}{v} - \frac{1}{v + 2}\right) dv = -\frac{dx}{x}$. Integrating: $\frac{1}{2}(\ln|v| - \ln|v + 2|) = -\ln|x| + \ln C_1 \implies \ln\left|\frac{v}{v+2}\right| = \ln\left|\frac{C}{x^2}\right|$. $\frac{v}{v+2} = \frac{C}{x^2}$. Substituting $v = y/x$: $\frac{y/x}{y/x + 2} = \frac{y}{2x + y} = \frac{C}{x^2} \implies x^2 y = C(2x + y)$.