In mathematics, an ordinary differential equation of the form
is called a Bernoulli equation when n≠1, 0, which is named after Jacob Bernoulli, who discussed it in 1695 Template:Harv. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.
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 .
Consider the Bernoulli equation (more specifically Riccati's 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 of . Integrating both sides results in the equations
The solution for is
as well as .
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