Find a first-order differential equation having the family y = C(x-1)e^x as integral curves

by RoRi

January 31, 2016

Find a first-order differential equation having the family

$y = C(x-1)e^x$

as integral curves.

Starting with the given equation we differentiate both sides with respect to $x$ ,

$y = C(x-1)e^x \quad \implies \quad y' = Ce^x + C(x-1)e^x = Ce^x + y.$

From the original equation we can solve for $C$ ,

$y = C(x-1)e^x \quad \implies \quad C = \frac{ye^{-x}}{x-1}.$

Therefore, we have the first-order differential equation

$y' = \frac{y}{x-1} + y \quad \implies \quad (x-1)y' - xy = 0,$

having the given family as integral curves.

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