The Black-Scholes partial differential equation treats the option price as a function of the underlying price and time . It relates the first derivative of in time, the first and second derivatives in price, and itself. Solving it from the payoff at expiry back to today gives the option price at each time and each price.
The Black-Scholes model assumes that the interest rate and the volatility are constant over the period. A European call has a price formula (an analytic solution), but in general the equation cannot be solved in closed form and is solved numerically.