Orthogonal Trajectories

Problem:

Find the orthogonal trajectories to the curves $y = x^2 + c$.

Solution:

First, we express the parabolas as differential equations

$\frac{dy}{dx} = 2x$

Next, the slope of the tangent lines for the orthogonal trajectories is determined, so we solve this

$\frac{dy}{dx} = \frac{-1}{2x}$

This is separable, and therefore the solution is simply

$y = \frac{-1}{2}sgn(x)\ln|x| + c$