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:
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19125.jpg)
If n = 1, the equation can also be written as a linear equation:
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19126.jpg)
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
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19128.jpg)
The equation above then becomes
which is linear in w (since n ≠ 1).
Example 1: Solve the equation
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19130.jpg)
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:
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19131.jpg)
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:
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19135.jpg)
Multiplying (**) through the
yields
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19136.jpg)
And an integration gives
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19137.jpg)
The final step is simply to undo the substitution w = y −2. The solution to the original differential equation is therefore
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/19138.jpg)