The full question is:

show that if $\displaystyle \frac{(N_{x} - M_{y})}{(x*M-y*N)}=R$ where R depends only on xy, then the differential equation M+N*y'=0 has an integrating factor of the form U(xy). Find a general formula for this integrating factor.

Using an equation from the text for U (for which finding a solution, U, means that the differential equation above is exact) I have:

$\displaystyle U= \frac{N*U_{x}-M*U_{y}}{M_{y}-N_{x}}$

then substituting in

$\displaystyle M_{y}-N_{x}=(yN-xM)*R$

and equating coefficients of N and M respectively,

$\displaystyle y*U*R=U_{x}$ and $\displaystyle x*U*R=U_{y}$

Is there some way that I can find the derivative of U wrt xy using the partial derivatives of U wrt x and y that I have above? Then I could integrate wrt xy to find U, right?