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

Theorem
The second order ODE:
 * $x^2 y'' = 2 x y' + \paren {y'}^2$

has the general solution:
 * $y = -\dfrac {x^2} 2 - C_1 x - {C_1}^2 \, \map \ln {x - C_1} + C_2$

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

Substitute $p$ for $y'$: