A path-dependent European option and the European strike average

Question
A PATH-DEPENDENT European option with expiry T is a European option
whose payoff is dependent on S(T) and the quantity
Z T
0
f(S(τ ), τ ) dτ
where f is a given function of S and t. By defined a new independent variable
I =
Z t
0
f(S(τ ), τ ) dτ
show that the stochastic differential equation satisfied by I is
dI = f(S,t)dt.
Use this result and an appropriate form of Ito’s lemma to show that the
partial differential equation satisfied by such options is
V − t + f(S,t)VI +
1
2
σ
2S
2VSS + rSVS − rV = 0.
Now consider the EUROPEAN AVERAGE STRIKE option where
f(S,t) = S(t).
Show that the partial differential equation is satisfied by solutions of the form
V = SU(η,t) (η = I/S)
provided that U satisfies a given partial differential equation (which you
should derive).