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Find an implicit formula satisfied by solutions of x dx + y dy = xy(x dy – y dx)

by RoRi

Assume solutions of the equation

    \[ x \, dx + y \, dy = xy (x \, dy - y \, dx) \]

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

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

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