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Find a first-order differential equation having the family arctan y + arcsin x = C as integral curves

by RoRi

Find a first-order differential equation having the family

    \[ \arctan y + \arcsin x = C \]

as integral curves.

We differentiate both sides of the given equation with respect to x,

    \begin{align*} &&\arctan y + \arcsin x &= C \\ \implies && \left( \frac{1}{1+y^2} \right) y' + \frac{1}{\sqrt{1-x^2}} &= 0 \\ \implies && \big( \sqrt{1-x^2} \big) y' + y^2 + 1 &= 0. \end{align*}

This is a first-order differential equation with the given family of curves as integral curves.

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