TD - Black-Scholes equation, a European down-and-out call and the payoff of a portfolio

Question
(a) Suppose that U(S,t) satisfies the Black-Scholes equation. Show that if
V is defined by
U(S,t) = S
nV (η,t)
where η = K/S and K and n are constant then
Vt +
1
2
σ
2Vηη + rηVη − rV = 0
provided n takes a particular value (which you should determine).
(b) A European DOWN-AND-OUT Call with strike E, expiry T and barrier
X is identical to a European Call option except for the fact that the
option cannot be exercised if the price of the underlying ever drops
below X. Explain briefly why the value D(S,t) of such an option must
satisfy D(X,t) = 0 and D(X, T) = max[S − E, 0]. Using the result
of part (a) or otherwise show that, if C(S,t) denotes the value of a
European Call option with strike E and expiry T, then
D(S,t) = C(S,t) − AS1−2r/σ2
C(K/S,t)
where A and K are constants (which you should determine).
(c) By considering the payoff of a portfolio which is long one down-and-out
Call and long one down-and-in Call, determine the value of a downand-in Call.