@mehran rajabi You can get some more roots like this:

Digits:= 15:
R:= table():
r:= 49.32883964:
Eq:= unapply(eq, x):
while r::numeric do 
    r:= RootFinding:-NextZero(Eq, r, maxdistance= 1e9);
    R[r]:= NULL
od:
indices(R, nolist, indexorder);
68.9303311172937, 106.974056704266, 151.062435474302, 198.382622867143, 245.339618384328,
291.310838331266, 338.964900033057, 387.685839558259, 434.873481747309, 481.556104974205,
672.896169599596, 721.880328695120, FAIL

The fact that FAIL was the last value returned doesn't necessarily mean that there are no more roots; it just means that NextZero gave up. If you want to search for more roots, let me know.

The following plot has 3 curves: red, blue, and green. The red one is your original linear interpolation. The green (vertical) corresponds to x = 808. The blue is another linear interpolation using the L[1] values in place of x[12]. Please explain the connection between this plot and what you actually want.

restart:
V[Ed13]:= <257.184, 230.4, 184.3, 138.2, 92.2, 46.1, 0>:
x[12]:= <-149, 208, 567, 925, 1283, 1642, 2000>:
L[1]:= <0, 358, 717, 1075, 1433, 1792, 2150>:
plot(
    [<x[12] | V[Ed13]>, < L[1] | V[Ed13] >, [[808,0], [808,250]]], 
    thickness= 2
);

@mmcdara Okay, it works in Maple 2019, but it's clunky unless you stabilize the extent of the vertical axis. That is, it helps if the vertical axis has the same "mathematical" length across the whole parameter range. You can do that by including a view option, or simply change the plot command to

plot(f, x= -3..3, 0..0.5)

Your worksheet is missing a definition of f. I assume that that just got deleted accidentally somehow, because it makes no sense (at least to me) without that definition.

We need to look at more-specific cases. Certainly it works in some cases, such as

asympt(HeunC(a, a, b, b, c, 1/z), z);

(works in Maple 2019).

@imparter 

Your first error above is that IBC is still not a set! It should be {IBC}.

The second error above is that you have Pa=0 where you should have Pa:= 0.

After you correct that, it'll complain that you don't have enough initial conditions. It wants 5. The only thing in your IBC that it considers an initial condition is p(r,-1) = 0.

The other post of mine that you linked to is not yet relevant! Your input hasn't even been accepted by pdsolve yet. Forget about the plots until the pdsolve is fixed,

Do not post followups as separate Questions!

You should start by entering the 5 equations, 5 solutions, and the intermediate variable definitions into Maple. Then, it should be very easy to verify that it is a solution. It may be a little more difficult to verify that it's the only solution.

You should change I and D to different variables because those are otherwise used by Maple.

@Kitonum The OP is referring to the kind of plot produced by Statistics:-ColumnGraph.

@Kitonum Based on the picture in the first version of this Question, I know that the intention is that k is a fixed number between 0 and n, inclusive.

@daljit97 I see that you've attempted to correct the Question. But "multiplication modulo 42" is not the same thing as "modular multiplication by 42". The former is the correct group operation. The latter isn't even a binary operation, let alone a group operation. 

"Modular division by 42" is not a well-defined group operation. A group operation has two operands from the group. You may be thinking of either "addition modulo 42" or "multiplication modulo 42", but those are very different groups.

@jgray07m I think that all that you need to do is keep the sliders for birth and death rates at 0. I don't think that dN/dt = 0 will be a problem.

@Joe Riel Thanks, Joe, for the Tally-once idea and for the profiling. I haven't looked at your code yet, but I'm sure that it will be enlightening. I wouldn't have guessed that Tally was the culprit, but it makes sense now. The reason that I count nonzeros rather than zeros is because that's all that the too-vaguely-named ArrayTools:-SearchArray does.

@Joe Riel Using expand will do more than using simplify in this and several related cases.

@Kitonum Yes, I like your interpretation.

Having taught 3-D Calculus with Analytic Geometry, my mental imagery of the problem involves lines and planes. Indeed, I was visualizing it as I fell asleep last night, and I put it into Maple when I awoke.