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

Theorem
The second order ODE:
 * $y y'' + \left({y'}\right)^2 - 2 y y' = 0$

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

Proof
Using Solution of Second Order Differential Equation with Missing Independent Variable:

After algebra, and reassigning constants:
 * $y^2 = C_2 e^{2 x} + C_1$