Second Order ODE/x y'' = y' + (y')^3

Theorem
The second order ODE:
 * $(1): \quad x y'' = y' + \paren {y'}^3$

has the solution:
 * $x^2 + \paren {y - C_2}^2 = C_1^2$

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

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