f=Log[Log[Log[x]]]

-x D[f, x,x]/f // Simplify

Y=First[y[x] /. DSolve[y''[x]+(x^α)y[x]==0 && y[1]==0 && y'[1]==1, y[x], x]]

y[x] /. DSolve[y''[x]+(x^(-2))y[x]==0 && y[1]==0 && y'[1]==1, y[x], x]

False && Rasterize[Plot3D[
Evaluate[y[x] /. DSolve[y''[x]+(x^α)y[x]==0 && y[1]==0 && y'[1]==1, y[x], x]],
{α, -3,0}, {x,1,10}
], ImageSize -> 500]

Evaluate[y[x] /. DSolve[y''[x]+(x^(-2))y[x]==0 && y[1]==0 && y'[1]==1, y[x], x]]

Y=First[y[x] /. DSolve[y''[x]+(x^α)y[x]==0 && y[1]==0 && y'[1]==1, y[x], x]];
ListPlot3D[
Table[Y,{α, -3,1,0.25}, {x,1,20,1}],
 PlotRange -> {-5,10}
]

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
], 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}]