The Bernoulli equation has the form

where $f$ and $g$ are continuous real functions and $k$ is a constant ($\neq 0$,  $\neq 1$).  Such a nonlinear equation is got e.g. in examining the motion of a body when the resistance of medium depends on the velocity $v$ as

The real function $y$ can be solved from (1) explicitly.  To do this, divide first both sides by $y^{k}$.  It yields

The substitution

transforms (2) into

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).