Integrating Factor for First Order ODE/Technique for finding Integrating Factor

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Proof Technique

Suppose you have a first order ODE which is in (or can be manipulated into) the form:

$M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$

and it is not homogeneous, exact or linear.

Then what you can do is evaluate:

$\dfrac {\partial M}{\partial y} - \dfrac {\partial N}{\partial x}$

and see what you get when you divide it by each of $N$, $M$, $N - M$ and $N y - M x$ in turn.

Then examine what you get to see if you have a function in $x$ only, $y$ only, $x + y$ or $xy$ respectively.

Suppose this has been achieved.

Then from Integrating Factor for First Order ODE, you have found an integrating factor and can solve the equation by using the technique defined in Solution to Exact Differential Equation.