The differential equation

where and , is known as the Bernoulli’s equation.

When is an integer, (1) has trivial solution .

To obtain nontrivial solution, we divide each term of (1) by to get,

Since ,

(2) can be expressed as

which is

.

Multiply throughout,

Let , (3) is transformed to a first order* linear* equation

,

giving the general solution of a Bernoulli’s equation (see Fig. 1)

Fig. 1

For a concrete example of Bernoulli’s equation, see “What moves fast, will slow down“