What is the Integrating Factor for x^n*y^m?

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Homework Help Overview

The discussion revolves around finding an integrating factor of the form x^n*y^m for a specific differential equation. The original poster seeks assistance in understanding the process to derive this integrating factor and solve the equation.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need for an example problem to clarify the context. The original poster provides a specific differential equation and requests guidance on finding the integrating factor. Another participant emphasizes the importance of actively engaging with the problem through algebraic manipulation.

Discussion Status

The conversation is ongoing, with participants exploring the necessary steps to derive the integrating factor. Some guidance has been offered regarding the manipulation of the equation and the conditions for it to be exact, but no consensus or resolution has been reached yet.

Contextual Notes

There is an emphasis on the need to derive two equations for m and n based on the equality of coefficients, which suggests that further exploration of these relationships is necessary. The original poster's request for help indicates a potential gap in understanding the underlying concepts.

bemigh
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Hey everyone,
I need to find an integrating factor of the form x^n*y^m, to solve a differential equation i have... however i do not know the process to solve for an integration of this form.. .any help??
Thanks
Steph
 
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can you give an example problem that your working on?
 
the problem is ( 12 + 5xy )dx + (6 (x/y)+ 3x^2)dy =0
and it says, find an integrating factor of the form (x^n) * (y^m), and solve the equation...
thanks sweetie
steph
 
RULE 1: Mathematics problems are not solved by staring at a problem until you remember the answer! They are solved by plugging things in and doing the algebra.
So TRY!

If you multiply the equation by x^ny^m you get
(12x^ny^m+ 5x^{n+1}y^{m+1})dx+ (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)dy= 0

In order for that to be an exact equation, you must have
(12x^ny^m+ 5x^{n+1}y^{m+1})_y= (6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x

Do the derivatives and see what m and n must be for those to be equal!

(12x^ny^m+ 5x^{n+1}y^{m+1})_y= 12mx^ny^{m-1}+ 5(m+1)x^{n+1}y^m
(6x^{n+1}y^{m-1}+ 3x^{n+2}y^m)_x= 6(n+1)x^ny^{m-1}+3(n+2)x^{n+1}y^m
Coefficients of the same powers must be equal. That gives two equations for m and n.
 
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