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

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

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

Proof
Using Solution of Second Order Differential Equation with Missing Independent Variable, $(1)$ can be expressed as:

$(2)$ is in the form:
 * $\dfrac {\mathrm d p}{\mathrm d y} + P \left({y}\right) p = Q \left({y}\right)$

where:
 * $P \left({y}\right) = \dfrac 1 y$
 * $Q \left({y}\right) = 2$

Thus:

Thus from Solution by Integrating Factor, $(2)$ can be rewritten as:

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