There are a few issues I can see with the formulation of your problem. First, as a set of ODE's you need initial conditions to completely pose the problem. Another issue is that to solve it numerically you need to know $f(S,T)$ (or at the very least be able to tell your computer to evaluate it). Both of these issues can be ignored if the solution is supposed to be analytical and general.
Since this is a homework and I am not sure what exactly you are being asked to do, I am not going to put any equations in this answer for the time being. I will tell you that it is possible to reduce your system to a single ODE with all of the unknowns eliminated except for $S$. The algebra to get there is not too difficult. Assuming stability the ODE can be solved with time-stepping such as Explicit Euler or an RK method. Once you have $S$ as a function of time, you can post-process for the rest of your unknowns.
If you let us know what type of class you received this in and what tools/methods you have learned you may get a bit more information.