Definitions

# Bernoulli differential equation

This topic in mathematics is named after Jakob Bernoulli. See Bernoulli's principle for an unrelated topic in fluid dynamics, named after the inventor Daniel Bernoulli.

In mathematics, an ordinary differential equation of the form

$y\text{'}+ P\left(x\right)y = Q\left(x\right)y^n,$

is called a Bernoulli differential equation or Bernoulli equation when n≠1, 0. Dividing by $y^n$ yields

$frac\left\{y\text{'}\right\}\left\{y^\left\{n\right\}\right\} + frac\left\{P\left(x\right)\right\}\left\{y^\left\{n-1\right\}\right\} = Q\left(x\right).$
A change of variables is made to transform into a linear first-order differential equation.
$w=frac\left\{1\right\}\left\{y^\left\{n-1\right\}\right\}$
$w\text{'}=frac\left\{\left(1-n\right)\right\}\left\{y^\left\{n\right\}\right\}y\text{'}$
$frac\left\{w\text{'}\right\}\left\{1-n\right\} + P\left(x\right)w = Q\left(x\right)$

The substituted equation can be solved using the integrating factor

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

## Example

Consider the Bernoulli equation
$y\text{'} - frac\left\{2y\right\}\left\{x\right\} = -x^2y^2$
Division by $y^2$ yields
$y\text{'}y^\left\{-2\right\} - frac\left\{2\right\}\left\{x\right\}y^\left\{-1\right\} = -x^2$
Changing variables gives the equations
$w = frac\left\{1\right\}\left\{y\right\}$
$w\text{'} = frac\left\{-y\text{'}\right\}\left\{y^2\right\}.$
$w\text{'} + frac\left\{2\right\}\left\{x\right\}w = x^2$
which can be solved using the integrating factor
$M\left(x\right)= e^\left\{2int frac\left\{1\right\}\left\{x\right\}dx\right\} = x^2.$
Multiplying by $M\left(x\right)$,
$w\text{'}x^2 + 2xw = x^4,,$
Note that left side is the derivative of $wx^2$. Integrating both sides results in the equations
$int \left(wx^2\right)\text{'} dx = int x^4 dx$
$wx^2 = frac\left\{1\right\}\left\{5\right\}x^5 + C$
$frac\left\{1\right\}\left\{y\right\}x^2 = frac\left\{1\right\}\left\{5\right\}x^5 + C$
The solution for $y$ is
$y = frac\left\{x^2\right\}\left\{frac\left\{1\right\}\left\{5\right\}x^5 + C\right\}$