Second Order ODE/y'' + 2 x (y')^2 = 0

Theorem
The second order ODE:
 * $(1): \quad y'' + 2 x \paren {y'}^2 = 0$

has the general solution:
 * $C_1 \, \map \arctan {C_1 x} = y + C_2$

Proof
The proof proceeds by using Solution of Second Order Differential Equation with Missing Dependent Variable.

Substitute $p$ for $y'$ in $(1)$ and rearranging: