TD - An up-and-out barrier put option and the Black-Scholes equation

Question
(a) An up-and-out barrier Put option is identical to a European Vanilla
Put, save for the fact that if any time during the life of the option
the asset price exceeds the barrier B, the option instantly becomes
(and remains) worthless. Assuming that the underlying asset pays no
dividends, explain briefly why the fair price V of the option must satisfy
the boundary value problem
Vt +
1
2
σ
2S
2VSS + rSVS − rV = 0, (S < B),
V (B,t) = 0, V (S, T) = max(E − S, 0), (S < B).
(Here as usual the asset value, strike price, volatility and interest rate
are denoted by S, E, σ and r respectively.)
(b) Assume now that for a particular up-and-out Put B¿E. Show that if
U(S,t) satisfies the Black-Scholes equation and V is defined by
U(S,t) = S
nV (η,t),
µ
η =
K
S
¶
where K is an arbitrary constant, then V also satisfies the Black-Scholes
equation provided n takes a specific value (which you should determine).
Hence or otherwise show that the fair value of and up-and-out barrier
Put is given by
V = PBS(S,t) −
µ
S
B
¶1−2r/σ2
PBS(B
2
/S,t)
where PBS denotes the value of a European Vanilla Put.