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

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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