Question

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.?

Answer 1

If there exists the I.F. `mu(w), w = x/y`, then:

`qquad {(mu_x = 1/ymu' ),(mu_y = - x /y^2mu'):} qquad " where "qquad mu' = (d mu)/(dw)`

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:

`qquad bbbA = bbbB`

`qquad mu ( M_y - N_x ) = (N/ y + M x/y^2 ) mu'`

`qquad mu y ^2 Delta = (y N + xM ) mu'`

`qquad:. ( y ^2 Delta)/(y N + xM ) = (mu')/mu qquad qquad = f(w)`

Therefore, for the I.F. `mu(w)` to exist, `( y ^2 Delta)/(y N + xM )` must be a function of `w`.