@Carl Love Exploring this just one step further indicates a bug in Maple 2020.1, Windows 10, June 10 2020 Build ID 1474787.

restart;
u:=1/3*ln(_C1^2 - 2*_C1*x + x^2 + 1) + 1/2*(-2*x + 2*_C1)*arctan(-x + _C1);
simplify(u);

I get
1/3*ln(_C1^2 - 2*_C1*x + x^2 + 1) + 1/3*(-3*x + 3*_C1)*arctan(-x + _C1)

which is correct, but couldn't possibly be intended.
######################

A simpler example:
 

restart;
u:=1/3*g(a) + (x+y)*f(b);
simplify(u);

Result:
1/3*(3*x + 3*y)*f(b) + 1/3*g(a)

PS. This goes all the way back to Maple 2016.2. It is not present in Maple 2015.2.

@Carl Love I need to use simplify directly on that term:
 

restart;
interface(version); # `Standard Worksheet Interface, Maple 2020.1, Windows 10, June 10 2020 Build ID 1474787`
my_sol:=y(x) = arctan(x - _C1)*x - arctan(x - _C1)*_C1 - ln(1 + (x - _C1)^2)/2;
simplify(my_sol); # 2 is still there
map(simplify,rhs(%)); # 2 is gone

@nm dsolve/numeric/bvp comes up with the initial profile n(x) = 0, which obviously is no good since n(x) is a factor in the denominator of the expression for n''(x).

So if 'Vortex' has an idea about the shape of a solution he could try giving an approximate solution in the form
approxsoln=[n(x) = f(x)]
where f(x) is something having that shape.

@AHSAN Since you don't say what it is that needs more explanation I shall make a few comments about what I think may require some elaboration. I will restrict myself to the last and fastest version.

1. The use of unapply. In your own worksheet you have the lines:
 

lambda1 := -3*(7*k^3*sigma^3 + 32*Q*k^2*sigma^2 - 11*k^2*sigma^3 + 54*Q^2*k*sigma - 44*Q*k*sigma^2 + 11*k*sigma^3 + 36*Q^3 - 54*Q^2*sigma + 32*Q*sigma^2 - 7*sigma^3)/(20*sigma^4);
## where sigma is given in terms of x earlier.
data1 := [seq([lambda1(x), x], x = 0 .. 0.6, 0.1)];

Since lambda1 is NOT defined as a function, but used as one anyway you may wonder why that works (it does!).
To see why, try this version instead:
 

data1 := [seq([lambda1, x], x = 0 .. 0.6, 0.1)];

Notice that you get exactly the same as before because the x inside lambda1 is given the different values of x = 0..0.6 with spacing 0.1.  The first version with lambda1(x) works because any number (e.g. -0.09786536940) will also work as the constant function with that value, thus -0.09786536940(x) = -0.09786536940 for any x.

Since I'm not going to use seq in my code I will turn lambda1 into an actual function of x. That is done by unapply.
After that I can do e.g. lambda1( 0.3) and expect to get what I intended.

2.  N is chosen so that the spacing between the x-values is 10^(-6). The vector V contains all the x-values and is given the datatype float (or explicitly float[8 ] ). Otherwise it would have been datatype=anything. To see that, try:
 

V:=Vector(7,i->0.6/6*(i-1));
op(3,V);

Why do I care about the datatype? Because for speed when V is big (you will have N = 600000) I would like to use hardware float computation (evalhf) and to get the most out of that I avoid data conversion from 'anything' to 'float[8]' by setting datatype=float[8] to begin with.
3. The use of map instead of the easier elementwise operation ~ .
Try

V:=Vector(7,i->0.6/6*(i-1),datatype=float[8]);
sin~(V); # OK
map(sin,V); # OK
evalhf(sin~(V)); # error
evalhf(map(sin,V)); # OK

4. Finally < W | V > creates a matrix with the columns W and V.

Is the vector you give named gln so that gln[i], i = 1..2 are its components?

@tomleslie dsolve, events works in Maple 12, thus also in Maple 13.

Your suggestion [[ gln[1]^2+gln[2]^2, halt]]  should work in principle, but I would like to see the actual system.
Will gln[1]^2+gln[2]^2 ever be found equal to zero, problem being that it is >=0 always?

The solution to that is the obvious: Replace gln[1]^2+gln[2]^2 by gln[1]^2+gln[2]^2-epsilon, where epsilon could be e.g. 1e-7.

There is a dog chasing jogger example in the help page for stop_cond in Maple 8. stop_cond has been superseded by events.
Dog_chasing_jogger_events.mw

For your worksheet to run without error you need kernelopts(floatPi=true).
This option exists in Maple 2015, but is not documented. It doesn't exist in earlier versions. I believe the default is false in 2015.
The option is documented in Maple 2016 and later versions and the default is true.
I found out since I have kernelopts(floatPi=false) in my maple.ini file.

If a univariate polynomial has a multiple root then that root is given as many times as its multiplicity.

I suppose that if in the solving process for you equation a univariate polynomial turns up this could create the situation you describe.

Try solving with infolevel[solve]:=3.
In the lines you will see polynomials mentioned. At the end a confusing message saying:
Main: Exiting solver returning 1 solution

@ogunmiloro Which Maple release/version are you using?
More importantly, what is your response to mmcdara's question about the fact that C__f has 110 members and 'times'  has only 20? Should the members in C__f be grouped in 20 groups of varying size and so how?

I haven't as yet found a description of the meaning of the procedure _pexports.

But it appears to be short for package exports and works this way:
 

M:=module() option package; export _pexports,a,b,c;
   _pexports:=proc() 
              [op({exports(M)} minus {':-c', ':-_pexports'})]
   end proc;
   a:=proc(x) 8*x end proc;
   b:=proc(x) x^2 end proc;
   c:=proc(x) sin(x) end proc;
end module;
##########
with(M); # Notice that only [a,b] is returned
a(7);
b(s);
c(8); # unevaluated
M:-c(8); # the long form works.

It most likely goes back to the introduction of modules in Maple (Maple 6 I believe). The code above certainly works in Maple 8 and later versions.
A neat facility which makes it possible to make a useful but "nerdy" procedure available to the programmer in other contexts than within M. I'm somewhat embarrassed that I haven't noticed this before (or maybe I have, but forgot).

@dharr Yes, this extra semicolon is allowed in Maple 2019 and 2020, but not before.
Trying in those two Maple versions:
p:=proc(x); local y; y:=x; y end proc;
we notice that the parser removes the semicolon.

Here is a variant where T is defined as a procedure with option remember and returning unevaluated if t is 0 or just the global name t..

restart;

ode2 := diff(varphi(t), t, t) + omega^2*sin(varphi(t));
p0 := evalf(10/180*Pi);
te:=6:
event2 := [[diff(varphi(t), t), T(t)=t]];
##
T:=proc(t) option remember; 
  if t::identical(':-t') or t=0 then 
    'T(t)' 
  else
     subs(ld2(t),T(':-t')) 
  end if
end proc;
##
ld2 := dsolve([eval(ode2, omega = 2*Pi), varphi(0) = p0, D(varphi)(0) = 0,T(0)=0], numeric,
               discrete_variables=[T(t)::float], events = event2,abserr=1e-12,relerr=1e-9);

op(4,eval(T)); # So far the only members of the remember table
plot(T(t),t=0..te); # Creates many members
{entries(op(4,eval(T)),nolist)} minus {T(0),T(t)};

The last line returns
{0., 0.5009535940730963, 1.0019071881466945, 1.5028607822207958, 2.0038143762954004, 2.504767970370508, 3.0057215644461186, 3.506675158522232, 4.0076287525988485, 4.508582346675968, 5.009535940753591, 5.510489534831716}

This one works as is in Maple 12 too.

I ran your interesting worksheet in Maple 2020.1.

It produced an error when coming to the statement

global liste_triangles:
The error was:

Error, global declaration unexpected outside procedure or module

Trying in a fresh worksheet in Maple 2015.2 and in Maple 2020.1 the followingt:
 

restart;
global liste_triangles;

I get the output _global(liste_triangles) in Maple 2015.2, but the error shown above in Maple 2020.1.

Just commenting out this attempt to declare liste_triangles makes your worksheet work fine in Maple 2020.1.

@brian bovril Since by gamma you most likely don't mean Euler's constant, you should either replace it by some other name or start your session with local gamma;  after restart.
Secondly, just replace gamma by beta or vice versa.
 

restart;
local gamma;
ode := x^2*diff(z(x), x, x) + (1 + gamma + beta)*x*diff(z(x), x) + gamma*beta*z(x) - cos(ln(x));
ode1:=eval(ode,gamma=beta);
sol:=dsolve({ode1, z(1) = 1, D(z)(1) = -1});
evalc(sol);

If x is known to be positive then this will shorten the answer considerably:
simplify(evalc(sol)) assuming x>0;

@Hnx I strongly recommend using worksheet mode and 1D math input (aka Maple input):

Go to Tools/Options/Display/Input display: Choose Maple Notation.

After that and without leaving the window that is open go to the menu Interface/Default format for new worksheets: Choose Worksheet.
Then press the button at the bottom labelled Apply Globally. Don't worry about the word "Globally"; these changes can just as easily be reverted.
Any new worksheet that you open will use the chosen format. Old worksheets or worksheets from other people won't change if brought into your copy of Maple.

Personally I never use 2D math input or document mode. I find it frustrating to work with as it seems to be for you.
Here is your code slightly modified (irrelevant though) in 1d-input and worksheet mode (and no equation labels: my personal preference).

MaplePrimes20-06-10_1D_worksheet.mw