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Sunday, December 27, 2015

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 $

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