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

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

has the solution:
 * $y = - \dfrac {x^2} 2 - C_1 x - C_1^2 \ln \left({x - C_1}\right) + C_2$

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

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