Here are a few hints.

(1) Each time you differentiate a polynomial, you reduce its degree by 1.

(2) If λ is an eigenvalue, with Dx = λx, then .

(3) If I understand this notation correctly, then v1(0) = constants, v2(0) = all linear functions, v3(0) = all quadratic functions. Their bases consist of monomial functions 1, x, x^2.