This is a theoretical question I am having trouble with.

(3) (Constant Coefficent Linear homogenous ODEs) Let p(z) = do + aix+ a2x2 + ...+ an-127 -1 + 27 be a polynomial with factorization p(x) = (x-71)(x-12) . .. (x -rn). Defined the associated differential operator as P ( D ) [ f] = ( ao + a , D + a 2 D 2 + . . . + an-IDn -1 + Dn) [f]. (a) Show that P(D) is a linear operator. (b) Show that P( D) [f] = (D - 71) (D - 12) ... ( D - In) [f]. (c) Show that ( D - TK ) ( D - r; ) [f] = (D - r; ) ( D - TK) [f]. (d) Show that if yx is a solution to the first order differential equation Dy = rky then yx is a solution to the equation P(D )y = 0 (e) Use the Fundamental Theorem of Algebra to conclude that the ODE P(D) [y] = 0 has at least as many solutions and it has distinct roots. (f) Find as many solutions as you can to the equation 1- y= 0