Integrating factor for a non-exact equation

Summary

Let $M\, dx + N\, dy = 0$ be an equation that is not exact on an open set $D$. An integrating factor for it is a nowhere zero differentiable function $\mu$ on $D$ such that $\mu M\, dx + \mu N\, dy = 0$ is exact. On a rectangle, with $M$, $N$ and $\mu$ of class $C^1$, the test for exactness makes this condition $(\mu M)_y = (\mu N)_x$. The two equations have the same solutions, because $\mu$ has no zero.

Remarks

If $(M_y - N_x)/N$ is a function $g(x)$ of $x$ only, then $\mu(x) = e^{\int g(x)\, dx}$ is an integrating factor. If $(N_x - M_y)/M$ is a function $h(y)$ of $y$ only, then $\mu(y) = e^{\int h(y)\, dy}$ is an integrating factor. The integrating factor of a linear first-order equation is the first case: $y' + p y - q = 0$ has $M = p y - q$, $N = 1$ and $(M_y - N_x)/N = p$.

Explanations

L1 human

Some equations are not exact, but they become exact after multiplication by a good function. In general, it is hard to find this function. In two common cases, the factor depends on one variable only, and a simple integral gives it.