how to define an example of ODE that represents an undamped forced oscillator whose time evolution is described by beats

Not really a mathematica problem but here is something to get you going in the Wolfram Language.

eqn = x''[t] + (2 π 100)^2 x[t] == f[t];
ic = {x'[0] == 0, x[0] == 0};
f[t_] := Sin[110 2 π t];
sol = NDSolve[Join[{eqn}, ic], {x}, {t, 0, 0.2}];
Plot[Evaluate[x[t] /. First[sol]], {t, 0, 0.2}]

Here I have made an oscillator with a natural frequency of 100 Hz and then applied a force with a frequency of 110 Hz.

The equation you are after is called eqn. The initial conditions are ic and the force has been defined as f[t].

You should be able to copy and paste that into Mathematica and get the plot I have shown.

Good luck