Home » Blog » Find the orthogonal trajectories of the family y2 = Cx

Find the orthogonal trajectories of the family y2 = Cx

by RoRi

Find the orthogonal trajectories of the family of curves by

    \[ y^2 = Cx. \]

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

    \[ y^2 = Cx \quad \implies \quad 2yy' = C. \]

Since C = \frac{y^2}{x} we then have,

    \[ 2yy' = \frac{y^2}{x} \quad \implies \quad y' = \frac{y}{2x}. \]

Therefore, the orthogonal trajectories satisfy the differential equation

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

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):