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

In mathematics, an ordinary differential equation of the form

<math>y'+ P(x)y = Q(x)y^n\,</math>

is called a Bernoulli differential equation or Bernoulli equation when n≠1. Dividing by <math>y^n</math> yields

<math>\frac{y'}{y^{n}} + \frac{P(x)}{y^{n-1}} = Q(x).</math>

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

<math>w=\frac{1}{y^{n-1}}</math>

<math>w'=\frac{(1-n)}{y^{n}}y'</math>

<math>\frac{w'}{1-n} + P(x)w = Q(x)</math>

The substituted equation can be solved using the integrating factor

<math>M(x)= e^{(1-n)\int P(x)dx}.</math>

[edit] Example

Consider the Bernoulli equation

<math>y' - \frac{2y}{x} = -x^2y^2</math>

Division by <math>y^2</math> yields

<math>y'y^{-2} - \frac{2}{x}y^{-1} = -x^2</math>

Changing variables gives the equations

<math>w = \frac{1}{y}</math>

<math>w' = \frac{-y'}{y^2}.</math>

<math>w' + \frac{2}{x}w = x^2</math>

which can be solved using the integrating factor

<math>M(x)= e^{2\int \frac{1}{x}dx} = x^2.</math>

Multiplying by <math>M(x)</math>,

<math>w'x^2 + 2xw = x^4,\,</math>

Note that left side is the derivative of <math>wx^2</math>. Integrating both sides results in the equations

<math>\int (wx^2)' dx = \int x^4 dx</math>

<math>wx^2 = \frac{1}{5}x^5 + C</math>

<math>\frac{1}{y}x^2 = \frac{1}{5}x^5 + C</math>

The final solution for <math>y</math> is

<math>y = \frac{x^2}{\frac{1}{5}x^5 + C}</math>

[edit] External links

• Bernoulli+ equation on PlanetMath
• Differential equation on PlanetMath
• Index of differential equations on PlanetMathbg:Диференциално уравнение на Бернули

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