>>11468744I was actually studying this yesterday. Basically we have that the function with the integrating factor is the true form of the differential equation matching df = df/dx dx + df/dy dy , but equality to 0 let's us factor the integrating factor out.
Suppose df/dx=h and df/dy = g, so we get hdx+gdy=0. Now suppose we can factor out a function r so that we have hdx+gdy= r(Mdx+Ndy)=0. If h and g are not zero then r is nonzero and the equation is equivalent to saying Mdx+Ndy=0. So the equation with the integrating factor is the real equation, but both describe the same set of curves.
