Even if I'm not familiar with this package, the Physics[Setup] help page says that vectordisplay should do the job:

vectordisplay   none, arrow, bold, or halfarrow.
When mathematicalnotation is set to true, the vectordisplay setting causes the Maple Standard GUI to display non-projected vectors with an arrow or halfarrow on top, or in bold, as specified.

An example (Maple 2015)

restart:
CountPrimes := proc(p::integer, q::integer)
  local k := 0:
  local i := nextprime(p):
  while i < q do
    k := k+1:
    i := nextprime(i):
  end do:
  return k
end proc:
      
start := 1:
for p in [seq(1+1000*k, k=0..9)] do
  printf("range %4d..%5d : nb of primes = %d\n", p, p+999, CountPrimes(p, p+999))
end do:

range    1.. 1000 : nb of primes = 168
range 1001.. 2000 : nb of primes = 135
range 2001.. 3000 : nb of primes = 127
range 3001.. 4000 : nb of primes = 119
range 4001.. 5000 : nb of primes = 118
range 5001.. 6000 : nb of primes = 114
range 6001.. 7000 : nb of primes = 117
range 7001.. 8000 : nb of primes = 106
range 8001.. 9000 : nb of primes = 110
range 9001..10000 : nb of primes = 111

Concerning the plot, here is a suggestion 

res := NULL:
start := 1:
for p in [seq(1+1000*k, k=0..9)] do
  res := res, plottools:-rectangle([p, 0], [p+999, CountPrimes(p, p+999)], transparency=0.7)
end do:
plots:-display(res)

To conclude:
"Can i make a procedure what calculates the latest mersenne prime number ?"  what does "latest" mean?
Advice: look at numtheory[mersenne] (Maple 2015, think it is numbertheory[mersenne] in more recent versions)

I believe you misinterpreted the phrase "This routine returns the "normalized" form of a linear differential equation, a differential operator list,...".

As written below it a differential operator list is not a list of odes but the list of the coefficients of the successive derivatives of the dependent variable of an ode.

As written in the DEnormal help page this list can be provided directly from the user, or derived from the ode by using convertAlg.
See the code bellow which is just a a commented version of the one given in this help page

restart:
with(DEtools):

# case of an ode
DE := 21*(-2*x^3)/(3*x^2-5*x+2)*y(x) + 42*(x^2)/(x+1)*x*(x-1)*D(y)(x) +
50*x^3/(x-1)^3*(D@@2)(y)(x) = x*sin(x);
DEnormal(DE,x,y(x));


# case of an operator list (here the operator list which corresponds to DE)
DE2 := convertAlg(DE,y(x));
DEnormal(DE2,x);

In the  convertAlg help page it's said that it "returns the coefficient list form for a linear ODE" (not of system of linear odes).
You can apply convertAlg and DEnormal for each ode in L this way

L := [diff(z(t), t, t)*m = 0, diff(x(t), t, t)*m - B*diff(y(t), t)*q = 0, diff(y(t), t, t)*m + B*diff(x(t), t)*q = 0] ;
V := [z(t), x(t), y(t)];
A := [seq(convertAlg(L[i], V[i]), i=1..3)];
N := DEnormal~(A, t);

and finally obtain the normal form of each ode this way

map(i -> add(N[i][1][j]*~(D@@(j-1))(op(0, V[i]))(t), j=1..nops(N[i][1])) = N[i][2], [$1..3]):
convert~(%, diff)
     [                                       / d      \    
     [                                     B |--- y(t)| q  
     [ d  / d      \       d  / d      \     \ dt     /    
     [--- |--- z(t)| = 0, --- |--- x(t)| = --------------, 
     [ dt \ dt     /       dt \ dt     /         m         

                            / d      \  ]
                          B |--- x(t)| q]
        d  / d      \       \ dt     /  ]
       --- |--- y(t)| = - --------------]
        dt \ dt     /           m       ]

But it's much simpler to do this

`Normal form of L`:= DEnormal~(L, t, V);

DEnormal.mw

Here is an example 

restart:

f := x -> x^2;
g := x -> x^3-1;
h := x -> abs(f(x)-g(x));

u := x -> ['x', 'f(x)', 'g(x)', 'h(x)'];

X := [seq(k, k = 1 .. 5)]:
M := convert([u(x), eval(u~(X))[]], Matrix);

# or maybe
M := convert([eval(u(x)), eval(u~(X))[]], Matrix);

# M can be saved using 
#    ExcelTools:-Export
#    ExportMatrix
#    ...
#
# For instance

n := cat(currentdir(), /"MyMatrix":
ExportMatrix(n, M);

Create_rable_mmcdara.mw

You could also use series(rhs(sol), x=0, 8) instead of asympt in your second example.

For your first example here is the Maple's solution based on a series expansion of the formal solution.
You will see it differs from MMA's from many points and I wasn't capable to transform th Maple's solution into MMA's one.

series.mw

But I guess you already did this yourself?

For the inverse Laplace transform of a function f(s) might exist, f(s) must vanish to 0 faster than
|s^n * f(s)|  (n strictly positive integer) as s tend to infinity.
This is not the case of d(s)

I don't say it's more efficient than what @tomleslie did, but I do believe it is more elegant and closer yo the formal problem:

 

Download The_probabilistic_way.mw

Obviously 7, 8, 12.. are othees among an infinity
 

restart:
sols := isolve(x^2+1=5*k):
S := op~(2, [sols]);

                 [x = 3 - 5 _Z1, x = 2 - 5 _Z1]

seq(op(eval(S, _Z1=n)), n=-10..0):
sort(rhs~([%]))
[2, 3, 7, 8, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 

  43, 47, 48, 52, 53]

Look to the attached file
proposal.mw

Like @Mac Dude I'm not sure I truly understand what you want.
Maybe something like this?
Maybe_This.mw

Here is a very simple version (continuity corrections do exist but are not implemented here).
Let E some event whose outcomes are 0 and 1

• n = sample size
• k = numer of times E=1 is observed amid these n observations
• p = the hypothetized probability that E=1
• alpha = the risk of first kind
• side = either "L" (left tail), "R", right tail or "LR" two-tailed (symmetric)
   

Download ExactBinomialTest.mw

I was writting the attached file as @dharr was posting his own answer.

This is basically the same thing, probably more verbose in my case
 

Download NonEuclideanLines_mmcdara.mw

Curiously the definitions of a vector basis seem to differ in English and in French.

In English a set F of vectors of some space E is a basis if every element of E can be written as a unique linear combination of elements of F.

In French a set ("family") F which verifies this condition is termed a "generating family".
A family F={u, v} is termed "free" if then a*u+b*v=0 ==> a=b=0.
And F is a basis iif it is a generating family and a free family. 

If the number of elements of F equals the dimension of V, then it's enough to veriffy that the family is a generating one or that it is a free one.
Generally it's easier to verify this second condition (in such a case the family is termed "maximally free"):

restart;
u := <1, 1>:
v := <1, 2>:

# Is {u,v} a free family?
cond := lambda*~u + mu*~v =~ <0, 0>:
solve(convert(cond, list), [lambda, mu])
                     [[lambda = 0, mu = 0]]

Correction is in 1D input mode
 

Download graph_mmcdara.mw

I used Maple 2015.2

Main problem; the value of delta is not set, I arbitrarily ghoosed delta:=1:

I changed theta[p] into theta__p (not sure this has any influence but I fell safer to use theta__p which is a variable which has noth1ng to do with theta).
The main point is that your equations 1 and 2 do not contain the functtions theta(eta) nor theta__p(eta).
This mean the inititial system can be spilt into 2 sub-systems:

• sys__12 made of eq1, eq2 and their bcs,
• sys__34 made of eq3, eq4 and their bcs.

Thus:

1. dsolve sys__12 (I use the option output=procedurelist)
    
2. Define a procedure K wich takes as argument the name eta or a numerical value of eta and returns in this case the evaluations of [f(eta), F(eta), diff(f(eta), eta), diff(F(eta), eta)] for this value of theta.
    
3. After having instanciate sys__34 with params, replace:
   1. f(eta) by K(eta)[1],
   2. F(eta) by K(eta)[2],
   3. diff(f(eta), eta) by K(eta)[3],
   4. diff(F(eta), eta) by K(eta)[4],
       
4. Solve the resulting sys__34 system.
   You must use the option known=K to tell Maple that K(eta)[1]...K(eta)[4] are known elsewhere.know
   
   Without any precaution you should get ab error which advices you tou use the midpoint submethod either midrich or middefer.
   Whatever the submethod I used the system seems so badly conditionned that I wasn't able to reach a reasonable accuracy (I desperately set abserr=1 to obtain a solution). This what I titled my answer "almost done".

Download MapleOde_mmcdara.mw