I've always wondered why does the differential equation notation for linear equations differ from the standard terminology of vector spaces.
We all know that the equation $y'' + p(x) y' + q(x)y = g(x)$ for some function $g$ is called linear and that the associated equation $y'' + p(x)y' + q(x) y = 0$ is called homogeneous. But why is that? WHY should mathematicians explicitly cause confusion with the rest of the theory of vector spaces?
What I mean by that is : Why not call the equation $y'' + p(x)y' + q(x)y = g(x)$ an affine equation and call $y' + p(x) y' + q(x) y = 0$ a linear equation? Because linear equations (in the sense of differential equations) are not linear in the sense of vector spaces unless they're homogeneous ; and linear equations (in the sense of differential equations) remind me more of a linear system of the form $Ax = b$ (which is called an affine equation in vector space theory) than of a linear equation at all.
Just so that I made myself clear ; I perfectly know the difference between linear equations in linear algebra and linear equations in differential equations theory ; I'm asking for some reason of "why the name".
Thanks in advance,