TD - Forward contract, Black-Scholes equation and an option on a future

Question
Give a short explanation of what is involved in a FORWARD CONTRACT
and explain briefly the major differences between a forward contract and a
future. Denoting the fair value of a forward contract by F, show that
F = S(t0)e
r(T −t0)
where t0 is the date on which the forward contract was greed, T is the delivery
date, r is the interest rate and S is the price of the underlying asset.
Now suppose that we wish to value a forward contract using the Black-Scholes
equation
Vt +
1
2
σ
2S
2VSS + rSVS − rV = 0.
Show that this equation has solutions of the form
V (S,t) + AS + Bf(t)
where A and B are constants and f is to be determined. By using this
solution along with the condition that must be satisfied at t = T show that
V (S,t) + S − Fe
−r(T −t)
and hence again show that
F = S(t0)e
r(T −t0)
.
An OPTION ON A FUTURE has a value V (F,t) where F = Se
−r(T −t)
.
Show from the Black-Scholes equation that V satisfies
Vt +
1
2
σ
2F
2VFF − rV = 0.