In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an ordinary differential equation

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

 of the form

is called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695 . Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.

Solution

Dividing by yields

A change of variables

Change of variables

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with new ones; the new and old variables being related in some specified way...

 is made to transform into a linear first-order differential equation.

The substituted equation can be solved using the integrating factor

Integrating factor

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...

Example

Consider the Bernoulli equation
We first notice that is a solution.
Division by yields
Changing variables gives the equations
which can be solved using the integrating factor
Multiplying by ,
Note that left side is the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of . Integrating both sides results in the equations
The solution for is
as well as .

Verifying using MATLAB symbolic toolbox by running

x = dsolve('Dy-2*y/x=-x^2*y^2','x')

gives both solutions:

0
x^2/(x^5/5 + C1)

also see a solution by WolframAlpha, where
the trivial solution is missing.