Linear First Order ODE/(1 + x^2) y' + 2 x y = 4 x^3

Theorem
The linear first order ODE:
 * $\paren {1 + x^2} \dfrac {\d y} {\d x} + 2 x y = 4 x^3$

has the general solution:
 * $y = \dfrac {x^4} {1 + x^2} + \dfrac C {1 + x^2}$

Proof
It is noticed that:
 * $\dfrac {\d} {\d x} \paren {1 + x^2} = 2 x$

and so $(1)$ can be rendered in the form: