Find the orthogonal trajectories for the family y = Ce^-2x

by RoRi

February 23, 2016

Find the orthogonal trajectories of the family of curves given by

$y = Ce^{-2x}.$

First, this family of curves satisfies the differential equation

$y = Ce^{-2x} \quad \implies \quad y' = -2Ce^{-2x}.$

Since $C = ye^{2x}$ we then have

$y' = -2y.$

Hence, the orthogonal trajectories satisfy the differential equation

$\begin{align*} y' = \frac{1}{2y} && \implies && 2yy' &= 1 \\ && \implies && 2 \int y \, dy & = \int dx \\ && \implies && y^2 &= x + C. \end{align*}$

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