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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
[/B]
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.