I am coursing differential equations and recently encountered with the concept of integrating factors. I have seen them to solve two types of ODEs: inexact and linear. A linear equation is an ODE in the form:

$\frac{dy}{dx}+p(x)y=q(y)$

the integrating factor ends up being:

$u(x)={e}^{\int p(x)dx}$

so that the equation comes to:

$\frac{d}{dx}(uy)=q(x)u(x)$

the equation can now be solved if $\int q(x)u(x)dx$ can be computed.

An inexact equation is an equation in the form

$A(x,y)dx+B(x,y)dy=0$

where

${A}_{y}\ne {B}_{x}$

(i.e. $Adx+Bdy$ is not an exact differential)

The integrating factor for these equations (I will call it $\mu $ for inexact equations) is a function such that

$(\mu A{)}_{y}=(\mu B{)}_{x}$

Expanding,

$\mu {A}_{y}+{\mu}_{y}A=\mu {B}_{x}+{\mu}_{x}A$

I have read the Wikipedia article, which says that to solve this equation where $\mu =\mu (x,y)$ requires partial differential equations, but if $\mu =\mu (x)$ or $\mu =\mu (y)$, then there is a straightforward formula for both, in terms of $A$ and $B$ (and their partial derivatives, respectively). But here is the important part: it says

"[...] in which case we only need to find $\mu $ with a first-order linear differential equation or a separable differential equation [...]"

Does this mean that this method can only be used for linear ODEs? In that case, I think the first method is way faster.