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Find the orthogonal trajectories of the family x2 – y2 = C

Find the orthogonal trajectories of the family of curves given by

    \[ x^2 - y^2 = C. \]


First, we find that the family of curves satisfies the differential equation

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

Therefore, the orthogonal trajectories satisfy the differential equation

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

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