y = Cos[Sqrt[2] Log[x]]

D[y, x]

x^3 D[y, {x, 3}] - 6 y

sol1[a_, b_, h_, n_] := 
      Module[{lst = {{0, a, b}}, s = {a, b}}, 
        Do[s = {{1, h}, {-h, 1}} . s // N; 
          lst = Append[lst, {N[k h], s[[1]], s[[2]]}], {k, 1, n}]; 
        lst]

sol1[0, 1, 4 Pi/10, 10]

Clear[ylist]

ylist[n_] := 
      Module[{lst, tbl}, lst = sol1[0, 1, 4 Pi/n, n]; 
        tbl = Table[{lst[[k, 1]], lst[[k, 2]]}, {k, 1, n + 1}]; 
        ListPlot[tbl, PlotJoined  ->  True]]

ylist[10]

ylist[50]

ylist[100]

ylist[150]

ylist[200]

Clear[ylist]

ylist[n_] := 
      Module[{lst, tbl}, lst = sol1[0, 1, 10 Pi/n, n]; 
        tbl = Table[{lst[[k, 1]], lst[[k, 2]]}, {k, 1, n + 1}]; 
        ListPlot[tbl, PlotJoined  ->  True]]

ylist[10]

ylist[20]

ylist[50]

ylist[150]

ylist[200]

sol2[a_, b_, h_, n_] := 
      Module[{lst = {{0, a, b}}, s = {a, b}}, 
        Do[s = {{1 - h^2/2, h}, {-h, 1 - h^2/2}} . s // N; 
          lst = Append[lst, {N[k h], s[[1]], s[[2]]}], {k, 1, n}]; 
        lst]

ylist2[n_] := 
      Module[{lst, tbl}, lst = sol2[0, 1, 4 Pi/n, n]; 
        tbl = Table[{lst[[k, 1]], lst[[k, 2]]}, {k, 1, n + 1}]; 
        ListPlot[tbl, PlotJoined  ->  True]]

ylist2[10]

ylist2[50]

ylist2[200]