Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.

Homogeneous: put $y = vx$. After substitution: $x\,dv/dx = \sqrt{1+v^2}$ ⇒ $dv/\sqrt{1+v^2} = dx/x$. Integrating: $\log(v + \sqrt{1+v^2}) = \log(cx)$ ⇒ $v + \sqrt{1+v^2} = cx$. Back-substitute $v = y/x$: $y/x + \sqrt{1 + y^2/x^2} = cx$, which simplifies to $y + \sqrt{x^2+y^2} = cx^2$.