Find a first-order differential equation having the family x² – y² = c as integral curves

by RoRi

January 31, 2016

Find a first-order differential equation having the family

$x^2 - y^2 = C$

as integral curves.

Since $x^2 - y^2 = C$ we can differentiate both sides with respect to $x$ to get

$x^2 - y^2 = C \quad \implies \quad 2x - 2yy' = 0 \quad \implies \quad yy' - x = 0.$

Thus, the first-order differential equation

$yy' - x = 0$

has the given family as integral curves.

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