Bernoulli differential equation

:See Bernoulli's equation for an unrelated topic in fluid dynamics.

Related Topics:
Bernoulli's equation - Fluid dynamics

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In mathematics, an ordinary differential equation of the form

Related Topics:
Mathematics - Ordinary differential equation

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:y'+ P(x)y = Q(x)y^n,

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is called a Bernoulli differential equation or often just Bernoulli equation.

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In order to solve this equation, we go through the following steps:

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First we divide by y^n:

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:rac{y'}{y^{n}} + rac{P(x)}{y^{n-1}} = Q(x). (1)

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Then, we change variables with

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:w=rac{1}{y^{n-1}}

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and

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:w'=rac{(1-n)}{y^{n}}.

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After substitution, we get the 1st order differential equation

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:rac{w'}{1-n} + (1-n)P(x)w = (1-n)Q(x) (2)

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which can be solved using the integrating factor

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:M(x)= e^{(1-n)int P(x)dx}.

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