Nterms = 5;
yy = Series[y[x], {x, 0, Nterms}];
equation = D[yy, x, x] - x*yy == 0;
initconds = {y[0] == AiryAi[0], y'[0] == AiryAiPrime[0]};
soln = SolveAlways[Join[{equation}, initconds], x];
truncatedSol = Normal[yy /. soln[[1]]]

Plot[{truncatedSol, AiryAi[x]}, {x, -6, 6}, PlotRange -> {-1, 1}]