Second Order ODE/y'' = 1 + (y')^2/Proof 1

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

Making the subtitution $u = A_1 e^y$:
 * $\mathrm d u = A_1 e^y \mathrm d y = u \mathrm d y$

Thus $(2)$ transforms into: