Actually, if the functions f(t), g(t), and h(t) have relatively standard Taylor series expansions, you could attempt a series solution. You cannot just plug in one valur of t in order to find y(t) at just that one point, for several reasons. 1. Merely by writing y'(t), you are writing a limit. Limits don't care what happens at a point, they care what happens near a point. Hence, you must have information near that point. 2. You have specified no initial conditions. The DE you wrote down, a linear second-order homogeneous ODE, has two arbitrary constants in its solution (due to the two integrations you are essentially performing) which must be determined by initial conditions. Therefore, there is no way to pin down the solution to the DE, even if you could exhibit it, because it would have two arbitrary constants.

Do you have a specific f(t), g(t), and h(t) in mind? If so, why not post them?