Bernoulli Equations

A differential equation of Bernoulli type is written as

displaymath49

This type of equation is solved via a substitution. Indeed, let tex2html_wrap_inline51 . Then easy calculations give

displaymath53

which implies

displaymath55

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:

(1)
Recognize that the differential equation is a Bernoulli equation. Then find the parameter n from the equation;
(2)
Write out the substitution tex2html_wrap_inline51 ;
(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:

displaymath55

(4)
Solve the new linear equation to find v;
(5)
Go back to the old function y through the substitution tex2html_wrap_inline59;
(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

displaymath83

Solution: Perform the following steps:

(1)
We have a Bernoulli equation with n=3;
(2)
Consider the new function tex2html_wrap_inline87 ;
(3)
The new equation satisfied by v is

displaymath91 ;

(4)
This is a linear equation:
4.1
the integrating factor is tex2html_wrap_inline93
4.2
we have tex2html_wrap_inline95
4.3
the general solution is given by

displaymath97

5
Back to the function y: we have tex2html_wrap_inline101 , which gives

displaymath103

6
All solutions are of the form

displaymath105

[Differential Equations]
[Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi
Last Update 6-23-98

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