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
We use the technique of formation of ordinary differential equation by elimination.

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

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

So:

Hence the result.