2.18.4 Third integrating factor

Using similar method If the above did not work, then we try

\[ R=\frac {1}{xM-yN}\left ( \frac {\partial N}{\partial x}-\frac {\partial M}{\partial y}\right ) \]

If \(R\) is function of \(t=xy\) only then the integrating factor is \(\mu =e^{\int Rdt}\) and let \(\overline {M}=\mu M,\overline {N}=\mu N\) then the ode \(\overline {M}\left ( x,y\right ) +\overline {N}\left ( x,y\right ) y^{\prime }=0\) is now exact.