Bernoulli Equations

A differential equation of Bernoulli type is written as


This type of equation is solved via a substitution. Indeed, let tex2html_wrap_inline51 . 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 tex2html_wrap_inline59 . 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:

Recognize that the differential equation is a Bernoulli equation. Then find the parameter n from the equation;
Write out the substitution tex2html_wrap_inline51 ;
Through easy differentiation, find the new equation satisfied by the new variable v.
You may want to remember the form of the new equation:


Solve the new linear equation to find v;
Go back to the old function y through the substitution tex2html_wrap_inline59;
If n > 1, add the solution y=0 to the ones you obtained in (4).
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:

We have a Bernoulli equation with n=3;
Consider the new function tex2html_wrap_inline87 ;
The new equation satisfied by v is

displaymath91 ;

This is a linear equation:
the integrating factor is tex2html_wrap_inline93
we have tex2html_wrap_inline95
the general solution is given by


Back to the function y: we have tex2html_wrap_inline101 , which gives


All solutions are of the form


[Differential Equations]
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Author: Mohamed Amine Khamsi
Last Update 6-23-98

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