# Bernoulli Equations

A differential equation of Bernoulli type is written as

This type of equation is solved via a substitution. Indeed, let . Then easy calculations give

which implies

This is a linear equation satisfied by the new variable v. Once it is solved, you will obtain the function . Note that if n > 1, then we have to add the solution y=0 to the solutions found via the technique described above.
Let us summarize the steps to follow:

(1)
Recognize that the differential equation is a Bernoulli equation. Then find the parameter n from the equation;
(2)
Write out the substitution ;
(3)
Through easy differentiation, find the new equation satisfied by the new variable v.
You may want to remember the form of the new equation:

(4)
Solve the new linear equation to find v;
(5)
Go back to the old function y through the substitution ;
(6)
If n > 1, add the solution y=0 to the ones you obtained in (4).
(7)
If you have an IVP, use the initial condition to find the particular solution.

Example: Find all the solutions for

Solution: Perform the following steps:

(1)
We have a Bernoulli equation with n=3;
(2)
Consider the new function ;
(3)
The new equation satisfied by v is

;

(4)
This is a linear equation:
4.1
the integrating factor is
4.2
we have
4.3
the general solution is given by

5
Back to the function y: we have , which gives

6
All solutions are of the form

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