From second order linear differential equation, two linearly independent equations can be found. Superposing the equations spans a plane of functions that serves as a the answer to the differential equation.
From this it led me to assume that then, the original differential equation must have belonged to a function space spanned by at least up to the 3rd derivative. Like how Hilbert space is a vector space analog to functions, span of two linearly independent answers among many other possibilities that do not satisfy the DE, would have to exist in space of higher order.
So, is this line of thinking correct? And if so, is there a limit up to which nth space the span of linearly independent functions can belong?