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Find a first-order differential equation having the family x2 + y2 + 2Cy = 1 as integral curves

by RoRi

Find a first-order differential equation having the family

    \[ x^2 + y^2 + 2Cy = 1 \]

as integral curves.

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

    \[ x^2 + y^2 + 2Cy = 1 \quad \implies \quad 2x + 2yy' + 2Cy' = 0. \]

From the given we equation we can solve for the constant,

    \[ x^2 + y^2 + 2Cy = 1 \quad \implies \quad C = \frac{1-x^2-y^2}{2y}. \]

Therefore we have

    \begin{align*} &&2x + 2yy' + 2Cy' &= 0 \\ \implies && x + yy' + y' \left( \frac{1-x^2-y^2}{2y} \right) &= 0 \\ \implies && y' (x^2-y^2-1) - 2xy &= 0. \end{align*}

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

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