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Find the orthogonal trajectories of the family x2y = C

Find the orthogonal trajectories of the family of curves given by

    \[ x^2y = C. \]


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

    \[ 2xy + x^2y' = 0 \quad \implies \quad y' = \frac{-2y}{x}. \]

Therefore, the orthogonal trajectories satisfy the differential equation

    \[ y' = \frac{x}{2y}. \]

Hence we have

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

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