Can solutions to the time-independent Schrodinger equation give solutions to the general equation if the potential is time dependent? I am reading through Griffiths Intro to Quantum, and he outlines how one can find solutions to the Schrodinger equation, by assuming the potential is constant in time (no time dependence), thus you can use separation of variables to get a time-independent equation, and find solutions to that equation as 'stationary states'. Then you use these stationary states in a linear combination to construct a general solution to the time dependent equation.
My question is, can you use these stationary states or some method of this form to construct a solution to the Schrodinger equation with a time-dependent potential (not constant in time). Griffiths says how his assumption in the book will be mostly just dealing with constant potentials, I want to know if you can use this method for non-constant potentials. If not, how do you solve it?
 A: The general answer to your final question,

How do you solve the Schrödinger equation for a generic time-dependent Hamiltonian?

is somewhere between "it depends", "it's complicated", "good luck", "with brute-force numerics", and "you don't".
But if the question is: are there commonly-used settings in which the solutions of the time-independent Schrödinger equation are useful for understanding the behaviour driven by a time-dependent Hamiltonian? then the answer is yes.
It happens very frequently that we have systems with Hamiltonians of the form
$$
\hat H(t) = \hat H_0 + \hat V_I(t),
$$
where $\hat H_0$ is (i) very complicated but (ii) time-independent, and $\hat V_I(t)$ is time-dependent but much simpler, and typically represents some form of interaction. One good example of such a setting is that of a many-electron atom driven by a laser field, where $\hat H_0$ is a complex multi-electron beast which involves lots of kinetic energies, spin-orbit couplings, and electrostatic interactions, plus a lot of correlation involved in its eigenstates, and where $\hat V_I(t)$ is just the dipole moment operator of each electron taken separately from the others, and multiplied by a simple sinusoidal function of time.
In this case, there is going to be a lot of hard work involved in solving for the eigenstates of $\hat H_0$, but this only needs to be done once, typically in advance. And once you have that, then you can have a much easier time in treating the effect that $\hat V_I(t)$ has in promoting transitions between the different eigenstates of $\hat H_0$.
If you want to see how this works in a general setting, then you should look up Time-Dependent Perturbation Theory in your favourite advanced-QM textbook.
Of course, it bears mentioning that this is not the only approach that fits the description you've given but that is too broad to provide an exhaustive list.
