Bernoulli differential equation


In mathematics, an ordinary differential equation of the form
is called a Bernoulli differential equation where is any real number other than 0 or 1. It is named after Jacob Bernoulli, who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli equation is the logistic differential equation.

Transformation to a linear differential equation

When, the differential equation is linear. When, it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For and, the substitution reduces any Bernoulli equation to a linear differential equation. For example, in the case, making the substitution in the differential equation produces the equation, which is a linear differential equation.

Solution

Let and
be a solution of the linear differential equation
Then we have that is a solution of
And for every such differential equation, for all we have as solution for.

Example

Consider the Bernoulli equation
.
The constant function is a solution.
Division by yields
Changing variables gives the equations
which can be solved using the integrating factor
Multiplying by,
The left side can be represented as the derivative of. Applying the chain rule and integrating both sides with respect to results in the equations
The solution for is