When a nonlinear first order ordinary differential equation has the form

with *n* ≠ 1, the change of variables

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

- Eliminating terms from higher order ODEs
- Period of a nonlinear pendulum
- Trading generalized derivatives for classical derivatives

[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.