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).