Orthogonal Trajectories/Examples/Exponential Functions

Theorem
Consider the one-parameter family of curves of graphs of the exponential function:
 * $(1): \quad y = c e^x$

Its family of orthogonal trajectories is given by the equation:
 * $y^2 = -2 x + c$


 * ExponentialsOrthogonalTrajectories.png

Proof
Differentiating $(1)$ $x$ gives:
 * $\dfrac {\rd y} {\rd x} = c e^x$

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $\dfrac {\rd y} {\rd x} = -\dfrac 1 y$

So:

Hence the result.