Say I have a general 1st-order homogeneous linear DE with constant coefficients.
$y'+ay=0$
The solution is $y=C_0e^{ax}$
Say I have the same thing but non-homogeneous.
$y'+ay=b$
The solution is $y=\frac{b}{a}+C_oe^{ax}$
The difference, obviously, is $\frac{b}{a}$. $\frac{b}{a}$ itself is a solution, and a constant solution at that. Bear with me, there's more.
So let's say that I don't have constant coefficients. Here's the homogeneous DE:
$y'+a(x)y=0$
The solution is $y=C_0e^{\int a(x)dx}$
And the non-homogeneous DE:
$y'+a(x)y=b(x)$
The solution being $y=e^{-\int a(x)dx} \int b(x)e^{\int a(x)dx}dx+C_0e^{\int a(x)dx}$
So the difference between the solutions is $e^{-\int a(x)dx} \int b(x)e^{\int a(x)dx}dx$.
So here's my question: Is there anything significant about each of the first terms in the solutions for both non-homogeneous solutions? They're both solutions themselves, but is there something particular about them, that separates them from all the other solutions?