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

by RoRi

January 31, 2016

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.

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