Bernoulli differential equation

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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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