we have and by the exact condition we have now and so put and solve that little ODE and you got your integrating factor.
The full question is:
show that if 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:
then substituting in
and equating coefficients of N and M respectively,
and
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?