Bernoulli equation

The Bernoulli equation has the form

${\displaystyle \frac{dy}{dx}}+f(x)y=g(x)y^k$ (1)

where $f$ and $g$ are continuous real functions and $k$ is a ($\ne 0$,  $\ne 1$).  Such a nonlinear equation (http://planetmath.org/DifferentialEquation) is got e.g. in examining the motion of a by $y^k$.  It yields

$y^{-k}{\displaystyle \frac{dy}{dx}}+f(x)y^{-k+1}=g(x).$ (2)

The substitution

$z:=y^{-k+1}$ (3)

transforms (2) into

$$\frac{dz}{dx}+(-k+1)f(x)z=(-k+1)g(x)$$

which is a linear differential equation of first order.  When one has obtained its general solution and made in this the substitution (3), then one has solved the Bernoulli equation (1).

Title	Bernoulli equation
Canonical name	BernoulliEquation
Date of creation	2013-03-22 15:15:03
Last modified on	2013-03-22 15:15:03
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Derivation
Classification	msc 34C05
Synonym	Bernoulli differential equation
Related topic	RiccatiEquation