I feel that the Answer by Markiyan and the comment by Tom have brought an unnecessary cloud of negativity over this situation. Yet it can be easily and simply solved by Maple when solutions exist.

Here's a worksheet with a procedure for the eigenvalues and a procedure for an associated eigenvector to each eigenvalue. I do an example over a field with a prime number of elements and one over a field with a prime power number of elements.

Download finite_field_eigenvalues.mw

The issue is exactly as Tom Leslie suggests in his fourth point: The lack of spacing in the expression P[B](p-w) causes Maple to interpret it as a function, P[B], whose argument is p-w. You no doubt intended for P[B] to be a coefficient to p-w. To express that multiplication, you need either a space or an explicit multiplication symbol between the two: either P[B] (p-w) or P[B]*(p-w).

This is only an issue when using Maple's odious 2D-input. In 1D-input, both f(x) and f (x) represent function application, and multiplication always requires an explicit symbol. I suggest switching to 1D-input.

The notation D(P[B])(p-w) means the derivative of the unknown (but known to be univariate) function P[B] evaluated at p-w.

I suspect that what you created was a table, not a list. I'd be able to tell instantly if I saw your code. So, try

display(convert(plotlist, list));

I suspect that you did something like the below, which creates a table, not a list:

for k to 5 do
     plotlist[k]:= plot(x^k, x= 0..1)
end do:

There's nothing wrong with the above code; indeed, one of the best ways to make a list is to first make a table and then convert it to list.

Yes, I can confirm it. To workaround, include the option linestyle= solid. This is also required if you want to see the wireframe or grid or mesh lines of surface plots.

Your input has been mangled horribly by Maple's 2D-input. In particular, most of your function applications have been misinterpretted as multiplications. For example, x(t) has been changed to x*t in several places. This is probably because you entered x (t) rather then x(t). I suggest that you use the 1D-input: Go to the Tools menu, select Options, then the Display tab. From the Input display pull-down, select Maple notation. Then select the Interface tab. From the "Default format for new worksheets" pull-down, select Worksheet. Then click on Apply Globally. Then open a new worksheet and retype your program.

Do you mean that you want the abcissa to be the circumference of a circle of radius R?

R:= 5:
plot(
     [
          [cos(t)*(cos(7*t)+R), sin(t)*(cos(7*t)+R), t= 0..2*Pi],
          [R*cos(t), R*sin(t), t= 0..2*Pi]
     ],
     color= [red, black], axes= none
);

plot3d([cos(t)*sin(p), sin(t)*sin(p), cos(p)], t= 0..2*Pi, p= 0..Pi);

You wrote:

I have another question: ... how could I find the values of root on vertical axis? In this figure I want to know the value of F(0) that cut the vertical axis?

Simply enter F(0), or in this case K(0), and you'll get the answer, 1. How is it possible that you are working with Kummer and Heun functions, and yet you don't know this basic math? It's called the y-intercept.

Yes, there is a shortcut. This can be done with a single call to dsolve with three parameters. Then the returned procedure is used for all n plots, resetting the parameters for each one. One parameter is the random value, and the other two are the starting values of t and L for the segment. This runs very fast if n is set to 36.

n:= 3:
Ra:= RandomTools:-Generate(float(range= 0.1e-1..0.5, method= uniform), makeproc):

b:= 0.1e-2:
T:= 10:

eq:= diff(L(t),t) = A*L(t)-b:
sol:= dsolve({L(T0) = L0, eq}, L(t), parameters= [T0, L0, A], numeric):

sol(0):= [L(t)=100]: #Load remember table with first initial condition.
for i to n do
     sol(parameters= [T*(i-1), eval(L(t), sol(T*(i-1))), Ra()]);
     p[i]:= plots:-odeplot(sol, [t,L(t)], t= T*(i-1)..T*i)
end do:

plots:-display(convert(p, list));

z:= Int(f(t-s), s= 0..1):
F:= unapply(subs(s= convert(s, `local`), z), t);

F(s);

There's no ambiguity in the above result: The two ss are distinct, and only the global one is accessible by name.

eval(%, s= q);

Try D(f)(-1) instead of diff.

Change

add((aver[1]-ma[j][2][1])^2, j = 1 .. 3)

to

add((rhs(aver[1])-rhs(ma[j][2][1]))^2, j = 1 .. 3)

Here are three solutions. My preference is strongly for the third of these, which avoids the need for any assign statements and any unevaluation quotes. Note that in the first solution, each variable name is surrounded by two pairs of single quotes, not one pair of double quotes.

Download Three_solutions.mw

Here are two ways that rely on the fact that op(0,f(t)) extracts f:

D~(map2(op, 0, x(t))) =~ 0:
(D@(x-> op(0,x)))~(x(t)) =~ 0:

The first method is more efficient.

Your code refers to signum in the line defining eq. The signum requires an argument---signum of what? In other words, it should be signum(...) with some algebraic expression in the parentheses. After you correct that, then proceed like this:

sol_2:= dsolve({cond, eq}, numeric, parameters= [X]);
sol_3:= proc(X)
     sol_2(parameters= [X]);
     eval(V(z), sol_2(3*l))
end proc;

Digits:= 4:
fsolve(sol_3);

I'm not sure about the precision control. You may need to adjust Digits or the error options of dsolve. I can't adjust it until you fix the signum.