Bernoulli differential equation

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

In mathematics, an ordinary differential equation of the form

[y'+ P(x)y = Q(x)y^n\,]

is called a Bernoulli differential equation or Bernoulli equation. Dividing by [y^n] yields

[\frac} + \frac} = Q(x).]

A change of variables is made to transform into a first-order differential equation.

[w=\frac}]

[w'=\frac}y']

[\frac + P(x)w = Q(x)]

The substituted equation can be solved using the integrating factor

[M(x)= e^.]

Example

Consider the Bernoulli equation

[y' - \frac = -x^2y^2]

Division by [y^2] yields

[y'y^ - \fracy^ = -x^2]

Changing variables gives the equations

[w = \frac]

[w' = \frac.]

[w' + \fracw = x^2]

which can be solved using the integrating factor

[M(x)= e^dx} = x^2.]

Multiplying by [M(x)],

[w'x^2 + 2xw = x^4,\,]

Note that left side is the derivative of [wx^2]. Integrating both sides results in the equations

[\int (wx^2)' dx = \int x^4 dx]

[wx^2 = \fracx^5 + C]

[\fracx^2 = \fracx^5 + C]

The final solution for [y] is

[y = \fracx^5 + C}]

External links

• [Bernoulli equation] on PlanetMath
• [Differential equation] on PlanetMath
• [Index of differential equations] on PlanetMath

 

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