Pensieve header: Plotting the solutions of the Airy equation, $y’’=xy$.

Ai1=NDSolve[y''[x]==x y[x] && y[0]==1 && y'[0]==0, y[x], {x, -10, 3}];
Ai2=NDSolve[y''[x]==x y[x] && y[0]==0 && y'[0]==1, y[x], {x, -10, 3}];
Ai=Join[Ai1, Ai2]

Plot[Evaluate[y[x] /. Ai], {x, -10, 3}]

a[0]=1; a[1]=0;a[2]=0; a[k_] /; k>2 := a[k] = ((a[k-3])/(k(k-1)));
Airy[n_] := (∑_(k=0)^n)a[k](x^k); Airy[18]

Plot[Evaluate[
Table[Airy[n], {n, {1,3,9,27,81}}]
], {x,-10,3}, PlotRange -> {-1.5,3.5}]