I want to solve a basic differential equation with a parameter and have the resulting solutions show as a set of functions in a merged plot. I want to be able to do two things:

1- Be able to compare the solutions for several different nearby values of the parameter. (For example to see which has a larger max.) This seems to restrict me to the usual two-dimensional plot (as opposed to allocating an extra axis for the parameter).

2- Have a fixed axes, as opposed to letting the scale to change.

Perhaps I need a manipulate as a subset of the main manipulate. For example let's consider

Manipulate[{Solns = NDSolve[{u''[t] + 5*u'[t] + 6*u[t] == 0, u[0] == 2, 
 u'[0] == a}, {u}, {t, 0, 5}], Plot[{u[t] /. Solns}, {t, 0, 5}]}, {a, 0, 5}]

Here as $a$ changes the trace of older functions are erased before newer ones are drawn, and the axes change. I am looking for a design that gives some presence to the older plots, and fixes the axes. For example I could have a slider $d$ whose values I set as $d=\delta (0,1,2,...,n)$, say $d=0.00, 0.01, 0.02, 0.03, ..., 0.10$, and then plot the solutions (using different colors?) for each of the values of $a-d$ for the current value of $a$.

Any suggestion is appreciated.

pndsv = ParametricNDSolveValue[{u''[t] + 5*u'[t] + 6*u[t] == 0, 
    u[0] == 2, u'[0] == b}, u, {t, 0, 5}, {b}];
delta = .3 Range[0, 10];

 Plot[Evaluate[Table[pndsv[b][t], {b, a - delta}]], {t, 0, 5}, 
  PlotRange -> {0, 3}, ImageSize -> 400, 
  PlotStyle -> (Opacity /@ Range[0, 1, .1])], {{a, 3}, 0, 5}]

enter image description here

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