Bernoulli differential equation

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Template:No footnotes 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.

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 (more specifically Riccati's equation).[1]

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 .

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}. Cited in Template:Harvtxt.

  • {{#invoke:citation/CS1|citation

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  1. y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013

External links