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.