I know that in order to solve: $\int y' y'' dx$
I can do $dy' = y''dx$, which is the same as subing $u=\frac{dy}{dx}$ and $du=\frac{d^2y}{dx^2} dx$ which results in:
$\int y' dy' = \frac {y'^2}{2} + C$
Why can't I do $dy = y'dx$ for:
$\int y'' dy = \int d^2y/dx^2 dy = \frac{d^2}{dx^2}(\int y dy) = \frac{d^2}{dx^2}(\frac {y^2}{2}) = y'^2+y y''$?