This differential equation is quite simple. So you can solve it analytically.
y[x_, \[Alpha]_] =
FullSimplify[
y[x] /. First@
DSolve[{\[Alpha]*y'''[x] + 8 y''[x] + 17 y'[x] + 10 y[x] == 0,
y[0] == 6*\[Alpha], y'[0] == -20, y''[0] == 84}, y, x],
0 >= x <= 1 && 1 <= \[Alpha] <= 2]
The is result not shown here.
And plot the DensityPlot
DensityPlot[y[x, \[Alpha]], {x, 0, 1}, {\[Alpha], 1, 2}]