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Find an implicit formula satisfied by solutions of (1 – x2)1/2 y′ + 1 + y2 = 0

by RoRi

Assume solutions of the equation

    \[ (1-x^2)^{\frac{1}{2}} y' + 1 + y^2 = 0 \]

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

    \begin{align*} (1-x^2)^{\frac{1}{2}} y' - (1+y^2) = 0 && \implies && \frac{-y'}{1+y^2} &= \frac{1}{\sqrt{1-x^2}} \\ && \implies && -\int \frac{1}{1+y^2} \, dy &= \int \frac{1}{\sqrt{1-x^2}} \, dx \\ && \implies && -\arctan y &= \arcsin x + C \\ && \implies && \arctan y + \arcsin x &= C. \end{align*}

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