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}]