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 ${y}^{\u2033}+p(x){y}^{\prime}+q(x)y=g(x)$ for some function g is called linear and that the associated equation ${y}^{\u2033}+p(x){y}^{\prime}+q(x)y=0$ 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 ${y}^{\u2033}+p(x){y}^{\prime}+q(x)y=g(x)$ an affine equation and call ${y}^{\prime}+p(x){y}^{\prime}+q(x)y=0$ 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".