Find a first-order differential equation having the family y = c (cos x) as integral curves

by RoRi

January 31, 2016

Find a first-order differential equation having the family

$y = C \cdot \cos x$

as integral curves.

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

$y = C \cdot \cos x \quad \implies \quad y' = -C \cdot \sin x.$

From the original equation we can also solve for the constant,

$y = C \cdot \cos x \quad \implies quad C = \frac{y}{\cos x}.$

Therefore,

$y' = -C \cdot \sin x \quad \implies \quad y' = -y \tan x \quad \implies quad y' + y \tan x = 0.$

This is a first-order differential equation with the given family of curves as integral curves.

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