RC Circuit with exponential input - Laplace
Yes, your analysis is correct. As below with a slightly simpler final expression

LT of the RHS of eq 1 is: Vo[1/s - T/(s+T)] = Vo/s(1+Ts)

LT of the LHS of eq 1 is: V(1+RCs)

Combining: V(1+RCs) = Vo/s(1+Ts), giving V = Vo/s(1+Ts)(1+RCs)

Partial fractions and inverse LT:

v(t) = Vo{1 + RC/(T-RC).e^(-t/RC) - T/(T-RC).e^(-t/T)}