No one was able to solve this question for me. If possible, could someone please walk me through

at least parts d, e, and f. If you could walk through

all of them, that would be nice, but definitely d, e, and f. Thank you!

V(3) (Constant Coefficent Linear homogenous ODEs) Let p(x) = do + aix + a2x2 + ... + an- 127-1+ x be a polynomial with factorization p(x) = (x-r1)(x -12) . .. (x -rn). Defined the associated differential operator as P ( D ) [ f ] = ( do + al D + a 2 D 2 + . . . + an-1 Dn-1 + Dn) [f]. V(a) Show that P(D) is a linear operator. V(b) Show that P(D) [f] = (D - 71) (D - 12) ... (D - In) [f]. (c) Show that ( D - TK ) ( D - r; ) [f] = (D - r; ) (D- TE) [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) ly] = 0 has at least as many solutions and it has distinct roots. (f) Find as many solutions as you can to the equation yin ) - y = 0