Let e be an nth order homogeneous linear differential equation. What does this mean? It means the derivatives of y are multiplied by various functions of x, and the highest derivative has order n, and the right hand side is equal to 0. Here is an example of order 2.
(x3+7x+5)y′′ - Exy′ + sin(x)y = 0
If f is a solution, then we can evaluate f(0), f′(0), f′′(0), and so on, through the first n-1 derivatives. This builds a vector in n space. Call this map m. Thus m(f) = v, mapping a solution function f to its derivatives at 0.
Since e is homogeneous, solutions can be added and scaled, hence the set of solutions forms a vector space. Furthermore, m respects addition and scaling. The derivatives of f+g are the derivatives of f plus the derivatives of g. Therefore m is a linear transformation from one vector space into another. The solution space of e has been mapped into n-space.
The previous theorem asserts existence and uniqueness for every vector of initial conditions. This is the inverse of the map described above. Given v, there is a unique f such that m(f) = v. Therefore m is onto (existence), and 1-1 (uniqueness). The set of solutions is isomorphic to n dimensional space.
If the right hand side of e is changed from 0 to r(x), so that e is no longer homogeneous, the solution set is a shifted vector space. Find one solution, h(x), and all solutions are of the form h+f, where f is a solution to the homogeneous equation. The dimension of the solution space is still n; we have simply shifted the entire space by adding h(x) to every function.