Hi Harashima,
It's used to sign off derivative formulae, subject to the relevant boundary conditions...
eg
1. I propose that f(St,t) = cos(St) - 4tK is the fair price of a call option. Check if this satisfies the PDE. It doesn't, it's nonsense, juts made it up. Chuck it in the bin.
2. I propose that f(St,t) = St - Kexp(-r(T-t)) is the fair price of a call option. Check if this satisfies the PDE. It does. You could check this, homework task ;-) However, in my formula, we have f(ST,T) = ST - K, which is not the correct BC for a call. Chuck it in the bin.
I give up :-(
3. Black and Scholes propose that f(St,t) = see page 47 with q=0 is the fair price of a call option. Check if this satisfies the PDE. It does. Check it gives the right BC, it does, f(ST,T) = max{ST - K,0}. Wahoo! We sign it off, "well done Black and Scholes, your proposal is correct. Here, have a Nobel prize (but only Scholes can have it because Black is dead)."
Good luck!
John