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

[bemigh]

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

 

[mathmike]

can you give an example problem that your working on?

 

[bemigh]

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

 

[HallsofIvy]

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.