TD - Black-Scholes equation and a down-and-out barrier call option

Question
Show that if V (S,t) is a solution of the Black-Scholes equation with zero
dividend yield, q = 0, then so too is S
αV (a/S,t), where a and α are constants,
provided α = 1 − 2r/σ2
.
A down-and-out barrier call option is an option which has the same payoff
as a call, but which becomes worthless if the spot price S ever drops below a
fixed barrier level B (even if it subsequently rises back above B, the option
remains worthless). Write down the Black-Scholes problem satisfied by the
barrier call.
Explain why you would expect the value of this option to be less than that
of an otherwise identical vanilla call option (without the barrier).
Assuming that the barrier lies below the strike, B < K, use the Black-Scholes
formula for a call value and the above observation to obtain an exact formula
for the barrier call option. [Hint, choose a above so that a/S = B when
S = B.] Confirm that the barrier option is less valuable than an otherwise
identical call option.