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
$$
\mu(y) = e^{\textstyle \int \left( \frac{\partial N / \partial x – \partial M / \partial y}{M} \right) \; dy}.
$$

E 2.5 Exercises