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:
the integrating factor ends up being:
so that the equation comes to:
the equation can now be solved if can be computed.
An inexact equation is an equation in the form
(i.e. is not an exact differential)
The integrating factor for these equations (I will call it for inexact equations) is a function such that
I have read the Wikipedia article, which says that to solve this equation where requires partial differential equations, but if or , then there is a straightforward formula for both, in terms of and (and their partial derivatives, respectively). But here is the important part: it says
"[...] in which case we only need to find 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.