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## Homework Statement

Find an integrating factor of the form ##x^Ay^B## and solve the equation.

##(2y^2-6xy)dx+(3xy-4x^2)dy=0##

## Homework Equations

##M=2y^2-6xy##

##N=3xy-4x^2##

##IF = exp(\int \frac{M_y-N_x}{N}\,dx)##

or

##IF = exp(\int \frac{N_x-M_y}{M}\,dy)##

## The Attempt at a Solution

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From the solutions in the back of my book I know that the integrating factor is supposed to be ##xy##, but I can't figure out how to find it.

First, I found the partial derivatives of M and N:

##M_y=4y-6x##

##N_x=3y-8x##

Since the two don't match, the equation isn't an exact equation.

To find the integrating factor, one of the latter two equations in section 2 must be a function of only a single variable, x for the first or y for the 2nd.

##\frac{M_y-N_x}{N}=\frac{4y-6x-3y+8x}{3xy-4x^2}=\frac{y+2x}{x(3y-4x)}##

I don't see any way to simplify this and get a single variable.

##\frac{N_x-M_y}{M}=\frac{3y-8x-4y+6x}{2y^2-6xy}=\frac{-y-2x}{2y(y-3x))}##

Again, I don't see any way to simplify this down to get one variable.

If neither of these can be simplified to a single variable I feel I'm at an impasse.