Working backward from the solution to the problem is useful in many areas. Euler, Lagrange, and other great masters did it. To get additional insight into the idea of integrating factors, start from a u(x, y) of your choice, find du = 0, destroy exactness by division by some F(x, y), and see what ODE's solvable by integrating factors you can get. Can you proceed systematically, beginning with the simplest F(x, y)?