Just use NDSolveValue to solve your problem:

    sol = ParametricNDSolveValue[{x''[t] +2 \[Delta]*x'[t] + \[Omega]0^2*x[t] +\[Alpha] x[t]^2 == 0,x[0] == 1, x'[0] == 0}, x, {t, t0, t1}, {\[Alpha]}]
    (*ParametricFunction[ <> ] *)
    Plot[Evaluate[Table[sol[\[Alpha]][t], {\[Alpha], {\[Alpha]1, \[Alpha]2,\[Alpha]3}}]], {t, t0, t1},PlotRange -> All]

[![enter image description here][1]][1]
    


  [1]: https://i.sstatic.net/UzZ92.jpg