Show that the equation M(x,y)dx+N(x,y)dy = 0 has an integrating factor which is a function of ratio of x and y.?
If there exists the I.F.
So, applying the I.F.:
# (M mu)_y = M_y mu - x /y^2 M mu' qquad bbbA#
# (N mu)_x = N_x mu + N 1/y mu' qquad bbbB#
For this to be an I.F., the mixed partials must be equal:
Therefore, for the I.F.