The Determination of Integrating Factor

From the differential equation

$M ~ dx + N ~ dy = 0$

Rule 1
If

$\dfrac{1}{N}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(x)$ , a function of x alone, then

$u = e^{\int f(x)~dx}$ is the integrating factor.

Rule 2
If

$\dfrac{1}{M}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(y)$ , a function of y alone, then

$u = e^{-\int f(y)~dy}$ is the integrating factor.

Note that the above criteria is of no use if the equation does not have an integrating factor that is a function of x alone or y alone.

Steps

Take the coefficient of dx as M and the coefficient of dy as N.
Evaluate ∂M/∂y and ∂N/∂x.
Take the difference ∂M/∂y - ∂N/∂x.
Divide the result of Step 3 by N. If the quotient is a function of x alone, use the integrating factor defined in Rule 1 above and proceed to Step 6. If the quotient is not a function of x alone, proceed to Step 5.
Divide the result of Step 3 by M. If the quotient is a function of y alone, use the integrating factor defined in Rule 2 above and proceed to Step 6. If the quotient is not a function of y alone, look for another method of solving the equation.
Multiply both sides of the given equation by the integrating factor u, the new equation which is uM dx + uN dy = 0 should be exact.
Solve the result of Step 6 by exact equation or by inspection.

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