Find the orthogonal trajectories of the family 2x + 3y = C

by RoRi

February 22, 2016

Find the orthogonal trajectories of the family of curves given by

$2x + 3y = C.$

From the given family of curves, we find a differential equation the curves all satisfy,

$2x + 3y = C \quad \implies \quad y' = -\frac{2}{3}.$

Letting $f(x,y) = -\frac{2}{3}$ , we know the orthogonal trajectories are the curves which satisfy a differential equation

$y' = -\frac{1}{f(x,y)} \quad \implies \quad y' = \frac{3}{2}.$

Therefore, the orthogonal trajectories are the curves,

$y = \int \frac{3}{2} \, dx = \frac{3}{2}x + C.$

Rearranging this (to get the answer Apostol gives in the back of the book) we have

$y = \frac{3}{2}x + C \quad \implies \quad 3x - 2y = C.$

Where $C$ is an arbitrary constant.

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