So let me get this straight: We can express any function we want by projecting it to that set of basis functions? But doesn't this mean that "solving" a particular equation is basically just a way of shoehorning our favourite function into a form that this particular differential equation is able to digest? Different differential equations have different sets of basis functions. But those sets of basis functions are not really "unique", i.e. one can combine them in some pattern and come up with a different set of basis functions that still satisfy the same differential equation. It's just a change of basis, a different point of view on the same function. Does it mean that a particular differential equation is more about that "point of view" (or how to construct it) than about the particular function that we're looking at?
I think you're right in that we COULD change the basis functions for a set differential equation and be able to reconstruct that equation; however, I don't think this is always useful. For example, in fourier analysis, we choose to construct a function based on complex exponentials of a frequency based on the original function because it lends itself well to solve for the coefficients. I'm sure like you said you could manipulate this basis, but then the ease of solving falls to the wayside.
