The differential equation

is known as **Bernoulli's equation.** If *n* = 0, Bernoulli's equation reduces immediately to the standard form first‐order linear equation:

If *n* = 1, the equation can also be written as a linear equation:

However, if *n* is not 0 or 1, then Bernoulli's equation is not linear. Nevertheless, it can be *transformed* into a linear equation by first multiplying through by *y* ^{− n },

and then introducing the substitutions

The equation above then becomes

which is linear in *w* (since *n* ≠ 1).

**Example 1:** Solve the equation

Note that this fits the form of the Bernoulli equation with *n* = 3. Therefore, the first step in solving it is to multiply through by *y* ^{− n }= *y* ^{−3}:

Now for the substitutions; the equations

transform (*) into

or, in standard form,

Notice that the substitutions were successful in transforming the Bernoulli equation into a linear equation (just as they were designed to be). To solve the resulting linear equation, first determine the integrating factor:

Multiplying (**) through the yields

And an integration gives

The final step is simply to undo the substitution *w* = *y* ^{−2}. The solution to the original differential equation is therefore