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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