# Integrating factor

1. Jan 31, 2006

### 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

2. Jan 31, 2006

### mathmike

can you give an example problem that your working on?

3. Jan 31, 2006

### 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

4. Feb 1, 2006

### HallsofIvy

Staff Emeritus
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.

Last edited: Feb 1, 2006