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

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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