First-Order Differential Equations: Separation Of Variables And Homogeneous Equations

Show that the differential equation

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

is homogeneous. Hence use the substitution ( v = \frac{y}{x} ) to solve the equation.

Use the method of separation of variables to solve the initial value problem

$$
\frac{dy}{dx}=y-y^2,\qquad y(2)=0.
$$

Give the solution in explicit form.