Nterms = 15;
yy = Series[y[x], {x, 0, Nterms}];
equation = D[yy, x, x] - x*D[yy, x] + Series[Sin[x], {x, 0, Nterms}] == 0;
initconds = {y[0] == 0, y'[0] == 1};
soln = SolveAlways[Join[{equation}, initconds], x];
truncatedSol = Normal[yy /. soln[[1]]]