Algebraic resolution of equations of the Black-Scholes type with arbitrary time-dependent parameters

We examine various forms of the Black-Scholes Equation using symmetry analysis and consider the standard problem of a terminal condition, namely u(T,x)=U. The models considered are the standard form of the equation to provide a benchmark, the nonhomogeneous form and two variations in which the parameters of the two basic models are explicitly time dependent. We find, subject to some constraints upon the parameters, a remarkable robustness in the results.