Applications of Bernoulli differential equations

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.

The post Applications of Bernoulli differential equations first appeared on John D. Cook.