Applications of Bernoulli differential equations

When a nonlinear first order ordinary differential equation has the form

\frac{dy}{dx} + P(x)\,y = Q(x)\, y^n

with n ≠ 1, the change of variables

u = y^{1-n}

turns the equation into a linear equation in u. The equation is known as Bernoulli’s equation, though Leibniz came up with the same technique. Apparently the history is complicated [1].

It’s nice that Bernoulli’s equation can be solve in closed form, but is it good for anything? Other than doing homework in a differential equations course, is there any reason you’d want to solve Bernoulli’s equation?

Why yes, yes there is. According to [1], Bernoulli’s equation is a generalization of a class of differential equations that came out of geometric problems.

Someone asked about applications of Bernoulli’s equation on Stack Exchange and got a couple interesting answers.

The first answer said that a Bernoulli equation with n = 3 comes up in modeling frictional forces. See also this post on drag forces.

The second answer links to a paper on Bernoulli memristors.

Related posts

[1] Adam E. Parker. Who Solved the Bernoulli Differential Equation and How Did They Do It? College Mathematics Journal, vol. 44, no. 2, March 2013.