(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.