If you put your code into a Code Sample (the first icon) it could be copied into Mathematica without having to type it in.
I suspect your code would work if you used f[i][t] for your functions instead of f[t,i].
For example:
NDSolve[{D[f[1][t], t] == f[2][t], D[f[2][t], t] == f[1][t],
f[1][0] == 1, f[2][0] == 1}, {f[1][t], f[2][t]}, {t, 0, 1}]
