Bernoulli differential equations are ordinary differential equations in the form

If or then it is linear. Otherwise it is non-linear, although they can be transformed into a first order linear differential equation by means of a substitution

This will then reduce the equation (after some manipulation and simplification) to a first order linear ODE which can be solved with an integrating factor.

For example:

Now we can make the substitution

This will give us:

This can now be solved the same way as a linear equation. The final answer is