If you need a symbolic approximation, there are a lot of things you can try:
For example, try to linearize the equation or a linear approximation to the problematic parts of the equation. If V[t] stays near 0, then then
Exp[B (Vg - V[t])]
Is reasonably well approximated by:
Normal@Series[Exp[B (Vg - V[t])], {V[t], 0, 1}]
E^(B Vg) - B E^(B Vg) V[t]
Substituing this in will give you an answer.
For more accurate approximations, try using the method of perturbations.
