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Find an implicit formula satisfied by solutions of xy(1 + x2) y′ – (1 + y2) = 0

by RoRi

Assume solutions of the equation

    \[ xy (1 + x^2) y' - (1+y^2) = 0 \]

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

    \begin{align*} xy (1+x^2) y' - (1+y^2) = 0 && \implies && y' \left( \frac{y}{1+y^2} \right) &= \frac{1}{x(1+x^2)} \\ && \implies && \int \frac{y}{1+y^2} \, dy &= \int \frac{1}{x(1+x^2)} \, dx \\ && \implies && \frac{1}{2} \log (1+y^2) &= \int \frac{1}{x} \,dx - \int \frac{x}{1+x^2} \, dx \\ && \implies && \frac{1}{2} \log (1+y^2) &= \log |x| - \frac{1}{2} \log (1+x^2) + C \\ && \implies && (1+y^2)(1+x^2) &= Cx^2. \end{align*}

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