Exact Linear Second-Order Equations

[Bruno Tolentino]

I found this page https://reference.wolfram.com/language/tutorial/DSolveExactLinearSecondOrderODEs.html and I was amazed, because I didn't see nothing like this. Exist theory about Exact Linear Second-Order Equations? What you know about?

 

[Jack Davies]

I think that it's just saying that, with that special condition on $a_i(x)$, the differential equation becomes:

$$a_0 y'' -a_0 '' y+a_1 y' + a_1 ' y =0$$

Which we can write as:

$$\frac{d}{dx}(a_0 y' -a_0' y) + \frac{d}{dx} (a_1 y) = 0$$

So with that very useful condition we can write it as a total derivative and then integrate up to a first order problem.