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Bernoulli differential equation - Wikipedia, the free encyclopedia

Bernoulli differential equation

From Wikipedia, the free encyclopedia

See Bernoulli's principle 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 when n≠1, 0. Dividing by yn yields

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

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

w=\frac{1}{y^{n-1}}
w'=\frac{(1-n)}{y^{n}}y'
\frac{w'}{1-n} + P(x)w = Q(x)

The substituted equation can be solved using the integrating factor

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

[edit] Example

Consider the Bernoulli equation

y' - \frac{2y}{x} = -x^2y^2

Division by y2 yields

y'y^{-2} - \frac{2}{x}y^{-1} = -x^2

Changing variables gives the equations

w = \frac{1}{y}
w' = \frac{-y'}{y^2}.
w' + \frac{2}{x}w = x^2

which can be solved using the integrating factor

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

Multiplying by M(x),

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

Note that left side is the derivative of wx2. Integrating both sides results in the equations

\int (wx^2)' dx = \int x^4 dx
wx^2 = \frac{1}{5}x^5 + C
\frac{1}{y}x^2 = \frac{1}{5}x^5 + C

The solution for y is

y = \frac{x^2}{\frac{1}{5}x^5 + C}

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