TD - Black-Scholes and call options

Question
(a) If A is a constant, show that V = AS is a solution of the Black-Scholes
equation. What “option” has this value?
Call options with strike E and expiry T are to be written on a share
that pays a dividend. The structure of the dividend payment is as
follows; a single payment with yield y (so that the amount received by
the holder is yS) will be made at a time td, T. The fair value of such
options is denoted by C(S, T; E, T).
(b) Explain why the option price remains continuous as the dividend date
is crossed, but the share price drops from S to (1−y)S. Give details of
the arbitrage possibilities that would exist if S did not jump to (1−y)S
across t = td.
(c) Let V (S,t; E, T) denote the fair (Black-Scholes) price for a Call option
on a share that pays no dividends with strike E and expiry T. Show
that
C(S,t; E, T) =
(
V (S,t; E, T) (td ≤ t ≤ T)
(1 − y)V (S,t; E/(1 − y), T) () ≤ t ≤ td)
(d) Using a financial argument or otherwise, determine whether
C(S,t; E, T) is larger or smaller than V (S,t; E, T).