Linear equations with varying coefficients

[V0ODO0CH1LD]

When talking about ordinary (real) linear differential equations with constant coefficients the idea is that we are dealing with the vector space of real functions over the field of real numbers. But when we allow the coefficients of a linear differential equation to be functions are we dealing with a different vector space (possibly one over the field of real functions) or are we dealing with the same vector space with some additional structure?

I guess the question is: for differential equations of the form
$$y''(x)+p(x)y'(x)+q(x)y(x)=f(x)$$
are the multiplications $p(x)y'(x)$ and $q(x)y(x)$ operations of the form $V\times{}V\rightarrow{}V{}$ (additional structure on the original vector field) or operations of the form $F\times{}V\rightarrow{}V{}$ (where $V$ and $F$ are the vector space and field, respectively)?

Also, if we are dealing with a vector space over the field of real functions what does the vector space consist of? And if we are dealing with the original vector space with some additional structure does it have a name?

Thanks!

 

[HallsofIvy]

You are not dealing with "a vector space over the field of real functions". You are dealing with a vector space over the field of real numbers, the members of the vector space being real functions. A vector space with the added operation of multiplication of vectors is "an algebra".