这个没有精确解,只能用数值解了,我给出的是一组解
range = 4 Pi;
ans = Table[
NDSolve[{y''[x] + Sin[x + k]*y[x] == x,y'[0] == 1,y[0] == k},
y[x],{x,0,range}],{k,0,3,0.1}];
Plot[Evaluate[y[x] /.ans],{x,0,range},PlotRange -> All]