A perpetual American put, the Black-Scholes equation and the optimal exercise price

Question
This question concerns the frequently-traded derivative product known as a
“Perpetual American Put”.
(a) Define the terms PERPETUAL, AMERICAN and PUT.
(b) Explain why, for a Perpetual American Put, the Black-Scholes equation
reduces to
1
2
σ
2S
2VSS + rSVS − rV = 0
where σ denotes volatility, r denotes interest rate, S denotes the asset
value and V denotes the value of the option.
(c) Given a simple arbitrage argument to show that the option value can
never be less than the early-exercise price and thus
V ≥ max(E − S, 0).
Explain why V must also satisfy V → 0 as S → ∞.
(d) Denote the optimal exercise price (i.e. the price which should automatically trigger exercise) by S
∗
. Explain why
(i) V (S
∗
) = E − S
∗
(ii) ∂V/∂S
∗ = 0
Hence or otherwise, show that the value of the option is given by
V (S) =
Ã
E
1 +
σ2
2r
!1+2r/σ2
σ
2
2r
S
−2r/σ2
.