Find a first-order differential equation having the family y² = Cx as integral curves

by RoRi

January 31, 2016

Find a first-order differential equation having the family

$y^2 = Cx$

as integral curves.

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

$y^2 = Cx \quad \implies \quad 2yy' = C.$

Then, since $y^2 = Cx$ we have $C = \frac{y^2}{x}$ . Substituting this on the right hand side we have the first-order differential equation

$2yy' = \frac{y^2}{x} \quad \implies \quad 2xy' - y= 0$

having the given family as integral curves.

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