Pensieve header: Little on the equation $y’’+x^\alpha y=0$.

Rasterize[ListPlot3D[Table[
Yα = First[y[x] /. NDSolve[y''[x]+(x^α)y[x]==0 && y[1]==0 && y'[1]==1, y[x], {x,1,20}]];
Table[Re[Yα], {x,1.,20,19/300}],
{α, -3.,1,4/300}
] // Transpose, DataRange -> {{-3,1}, {1,20}}], ImageSize -> 500]

Plot[Evaluate[Table[
y[x] /. DSolve[y''[x]+(x^α)y[x]==0 && y[1]==0 && y'[1]==1, y[x], x],
{α,-3,1,0.25}]], {x,1,26}]