Show that the orthogonal trajectories for the family of curves $\displaystyle y = ae^{-x}$ (a constant) are a family of parabolas.

Attempt:

$\displaystyle y = ae^{-x}$

$\displaystyle \frac{dy}{dx} = -ae^{-x}$

$\displaystyle \frac{dy}{dx} = \frac{1}{ae^{-x}}$

$\displaystyle \int dy = \int \frac{1}{ae^{-x}}dx$

$\displaystyle y = \frac{1}{a}\int e^{x}dx$

$\displaystyle y = \frac{1}{a}e^{x} + C$

but the correct answer was:

$\displaystyle \frac{1}{2}y^2 = x +C$

I cant seem to see where I went wrong though.