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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.

Find a first-order differential equation having the family y = Ce-2x as integral curves

Find a first-order differential equation having the family

    \[ y = Ce^{-2x} \]

as integral curves.


Since y = Ce^{-2x} we can differentiate both sides with respect to x to get

    \[ y = Ce^{-2x} \quad \implies \quad y' = -2Ce^{-2x}. \]

Then, we plug in our original equation Ce^{-2x} = y on the right hand side to get the first-order differential equation

    \[ y' = -2y \quad \implies \quad y' + 2y = 0. \]

having the given family as integral curves.