2.5 | Special Integrating Factors

If the equation
$$M(x,y) \; dx + N(x,y) \; dy = 0$$
is not exact, but the equation
$$\mu(x,y)M(x,y)\;dx + \mu(x,y) N(x,y) \; dy = 0,$$
which results from multiplying the first equation by the function $$\mu(x,y)$$, is exact, then $$\mu(x,y)$$ is called an \textbf{integrating factor.}
If $$(\partial M / \partial y – \partial N / \partial x)/N$$ is continuous and depends only on $$x,$$ then
$$\mu(x) = e^{\textstyle \int \left( \frac{\partial M / \partial y – \partial N / \partial x}{N} \right) \; dx}$$
is an integrating factor for $$M \; dx + N \; dy = 0.$$
If $$(\partial M / \partial y – \partial N / \partial x)/N$$ is continuous and depends only on $$y,$$ then
$$\mu(y) = e^{\textstyle \int \left( \frac{\partial N / \partial x – \partial M / \partial y}{M} \right) \; dy}$$
is an integrating factor for $$M \; dx + N \; dy = 0.$$
Method for Finding Special Integrating Factors
If $$M \; dx + N \; dy = 0$$ is neither separable no linear, compute $$\partial M/ \partial y$$ and $$\partial N / \partial x.$$ If $$\partial M / \partial y = \partial N / \partial x,$$ then the equation is exact. If it is not exact, consider
$$\frac{\partial M / \partial y – \partial N / \partial x}{N}.$$
If this is a function of $$x$$, then an integrating factor is given by
$$\mu(x) = e^{\textstyle \int \left( \frac{\partial M / \partial y – \partial N / \partial x}{N} \right) \; dx}.$$

If not, consider
$$\frac{\partial N / \partial x – \partial M / \partial y}{M}.$$
If this is a function of $$y$$, then an integrating factor is given by
$$\mu(y) = e^{\textstyle \int \left( \frac{\partial N / \partial x – \partial M / \partial y}{M} \right) \; dy}.$$