@Christopher2222 Yes, in the last two weeks or so, I have noticed a slow down of the speed with which typing is accepted. 

@abiodunowoyemi9 The problem that you just posted is exactly the same as the original.

@jud The help page ?GroupTheory,Subgroup that I'm looking at says that the first argument to Subgroup is a "list or set of generators for the subgroup." The basic list constructor is [...], and the basic set constructor is {...}. Sets are automatically sorted and simplified to remove repeated elements. The problem with using {RandomElement(G), RandomElement(G)} is that that expression automatically simplifies to a single element before the random elements are chosen. This phenomenon can be observed in a much simpler setting, and it should be kept in mind whenever making multiple random selections with a single command:

restart:
{rand(), rand()};
                         {395718860534}
restart:
[rand(),rand()];
                  [395718860534, 193139816415]
 

@jalal One way to animate is the viewpoint option. For example, you can add viewpoint= "circleleft" to any of the final display commands. There are a vast number of suboptions for finer control. See the help page ?plot3d,viewpoint for details.

I just realized that my Answer may give the impression that I am recommending that you use solve instead of fsolve. I am not. I recommend that you fix your calls to fsolve so that functions are not used as variables. Note that indexed names (such as y[1]) work fine as fsolve variables, and they're practically no different than functions.

@abiodunowoyemi9 It's a version issue. You should use the pull-down list in the Question header to state which Maple version you are using. Below, I've removed some of the newer features from my code. Please try this:
 

Download UndeterminedCoeffs.mw

@vv Yes, I noted that and replied to him above.

@tomleslie You have exactly the same typos on the right sides of the 3rd, 4th, and 5th equations as VV has. I guess he used your encoding of the equations.

@vv You have a few typos on the right-sides of the 3rd, 4th, and 5th equations. I've corrected these below. Running the corrected code (in Maple 2019), I experience the same thing that I said above: an undetermined very long time for the 6th solution.

restart;
sys:= [ 
[diff(v__0(t),t$2)+v__0(t)=0,  v__0(0)=1, D(v__0)(0)=0],
[diff(v__1(t),t$2)+v__1(t)=-v__0(t)^3,  v__1(0)=0,  D(v__1)(0)=0],
[diff(v__2(t),t$2)+v__2(t)=-3*v__0(t)^2*v__1(t),  v__2(0)=0,  D(v__2)(0)=0],
[diff(v__3(t),t$2)+v__3(t)=-3*v__0(t)*v__1(t)^2-3*v__0(t)^2*v__2(t),  v__3(0)=0, D(v__3)(0)=0],
[diff(v__4(t),t$2)+v__4(t)=-6*v__0(t)*v__1(t)*v__2(t)-3*v__0(t)^2*v__3(t)-v__1(t)^3,  v__4(0)=0,  D(v__4)(0)=0],
[diff(v__5(t),t$2)+v__5(t)=-6*v__0(t)*v__1(t)*v__3(t)-3*v__1(t)^2*v__2(t)-3*v__0(t)^2*v__4(t)-3*v__0(t)*v__2(t)^2,  v__5(0)=0, D(v__5)(0)=0]
]:
print~(sys);
s:=NULL:
for i to 6 do  s:=s, dsolve(eval(sys[i], [s])); print([s][-1]) od:

@tomleslie It was not my intention to restrict n to integers. Yes, if that had been my intention, I would've included adaptive= false. My intention was to make sure that every integer was included along with whatever other points plot decided to include on its own.

Yes, I understand your point about the inner floors. They aren't needed in my Answer. I mistakenly thought that I was using the OP's original function, but I was actually using your modification of it due to a hasty copy and paste. Sorry about that. So, the code of my Answer should be changed to 

f:= n-> ceil(sqrt(4*n)) - floor(sqrt(2*n)) - 1:
plot(f, 10...100, sample= [$10..100]);

The distance function that I gave before should be modified to account for cases that require no horizontal travel. This doesn't affect the final results previously obtained because your example doesn't have any such pairs of points. The new distance function is 

dist:= proc(p::[integer,integer], q::[integer,integer])
local H;
    if p[1]=q[1] then abs(p[2]-q[2])
    else min((add@abs~)~([seq]([p[2]-H, p[1]-q[1], H-q[2]], H= [1,10])))
    fi
end proc      
:

@jalal In the line defining fnew, try changing ^+ to ^%T. They mean the same thing---transpose the matrix---but ^+ only works in (plaintext) 1D input.

@Teep Here is the code to highlight with arrows the direction of travel for the solution to the traveling salesman problem.

restart:
GT:= GraphTheory: Pl:= plots: Co:= combinat:
dist:= proc(p::[integer,integer], q::[integer,integer])
local H;
    min((add@abs~)~([seq]([p[2]-H, p[1]-q[1], H-q[2]], H= [1,10])))
end proc      
:
HighlightTour:= proc(    
    #edges:
    E::set([And(set({name, string, integer}), 2 &under nops), numeric]),
    #tour:
    T::list({name, string, integer}), #the  tour
    #vertices (optional, for positions only):
    V::list({name, string, integer} = [realcons,realcons]):= 0
)
uses GT= GraphTheory;
local G0,Gt;
    (G0,Gt):= GT:-Graph~(
        [selectremove](e-> e[1]::set, subs((`{}`=`[]`)~(T[..-2],T[2..]), E)),
        `if`(V=0, [][], 'vertexpositions'= rhs~(V))
    )[];
    GT:-HighlightTrail(Gt, T, _rest, 'inplace');
    G0,Gt
end proc
:
pts2 := [A = [5, 0], B = [8, 9], C = [2, 8], D = [4, 3], E = [7, 4]]:
Edgs:= {seq}([({op}@lhs~)(e), dist(rhs~(e)[])], e= Co:-choose(pts2, 2)):
G1:= GT:-Graph(Edgs):
(distance, Tour):= GT:-TravelingSalesman(G1);
            distance, Tour := 38, [A, D, C, B, E, A]
(G0,Gt):= HighlightTour(Edgs, Tour, pts2);
G0, Gt := Graph 1: an undirected weighted graph with 5 vertices 
   and 5 edge(s), 
  Graph 2: a directed weighted graph with 5 vertices and 5 arc(s)

Pl:-display(GT:-DrawGraph~([G0,Gt], style= circle, stylesheet= "legacy"));

#Using actual vertex positions:
Pl:-display(GT:-DrawGraph~([G0,Gt], stylesheet= "legacy"));

To see the tour (red) edges only, remove G0 from either of those plot commands.
 

I'm having some trouble at the moment loading the workbook format (*.maple). Would it be possible for you to duplicate the error in the worsheet (*.mw) format?

If you don't give any constraints, then obviously there's no upper bound. For example, you could make A1 any large value and make all the other variables 0.