@acer sort may fail when the detected indets are too far from polynomials. E.g.
sol := y(x) = _C1*x*sin(2*ln(x))+_C2*x*cos(2*ln(x))/(x+sin(x));
 

@acer Yes, I know, but the probability is low.
I saw your solution in the other thread, but it also has some issues (see there).

Edit. A workaround is to declare c as local.
BTW, you meant ex1a:=...  So,

restart;
ex1:=1/cosh(x)+   (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x)*_C1 + x*exp(_C3+x+_C2)*C:
ex2:=_C1/cosh(x)+ (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x):
ex3:=_C1/cosh(x)+ (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x)*_C2:
ex1a:=1/cosh(x)+   (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x)*c[1] + x*exp(c[3]+x+c[2])*C:
ex2a:=c[1]/cosh(x)+ (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x):
ex3a:=c[1]/cosh(x)+ (cosh(x)^2+cosh(x)*sinh(x)+x)/cosh(x)*c[2]:
Cc:=proc(u)
  local c, Cc1 := C -> c[parse(convert(C,string)[3..])];
  subsindets(u, `*`,
    proc(u)
      local k,r;
      k,r:=selectremove(type, [op(u)], suffixed(_C,integer));
      if nops(k)<>1 then return u fi;
      `*`(Cc1(k[]),r[])
    end):
  subsindets(%, suffixed(_C,integer), Cc1)
end:
Cc(ex1); Cc(ex2); Cc(ex3);

@ecterrab `print/_C`   is applied (in the original version) for _C()  and works (for display).
See my comment above for Latex(_C()...), and my solution above to place c[k] in front of a product.

@nm There are a few problems here.

1. The subs in the original version must be used only for display purposes.

2. Latex fails  for  _Cn()
Latex(_C4()/x+x);   ==> \frac{1}{x}+x
Latex(_C4()+x);      ==> \mathit{\_C4} {}+x

3. It seems that in order to put c[k] instead of _Ck in front of a product, 
sort is not always useful, so, it will be necessary to use subsindets. 
 

@nm I think that this works.

assign(seq(`print/_C`||k, k=0..10) = seq(c[k], k=0..10)):
S:=[seq(_C||k = _C||k(), k=0..10)]:

sol:=dsolve(t^2*diff(diff(y(t),t),t)-t*diff(y(t),t)+5*y(t) = 0,y(t)):
subs(S, sol);

@JAMET Like this:

qq:=expand(q(3,b)); # a = arbitrary (a=3 here)
Delta:=diff(qq,x,y)^2-diff(qq,x,x)*diff(qq,y,y);
solve(Delta>0); 

Be aware that an ellipse could be complex, so an extended analysis is needed.
Why don't you try something yourself in Maple and show the results?

@JAMET In the context of my answer,

qq:=q(6,-6):
Delta:=diff(qq,x,y)^2-diff(qq,x,x)*diff(qq,y,y);

==> Delta := -108144 ==> ellipse

@Carl Love I agree in general, but not in this case. 

@JAMET 
1. Use the equations of the lines passing thru each pair of points.
2. You must find the equation of the conic first. Then you may use the geometry package.

@emendes Just curious: do you really need all the computations done in advance? Or is it a CPU test?
It is not difficult to write a compilable version, but actually Maple in not needed for this problem, so why not use e.g. C?

@Katatonia 

SYS1:=u>=0, r^2 - 2*r*c -r +2*c = u^2, -r+c+1+u>0,  0<=c, c<=1 , 0<r, r<1, r<2*c:
SolveTools:-SemiAlgebraic([SYS1], [r,u], parameters=[c]);

(Here, of course, the auxiliary variable u can be ignored when reading the answer.)

@s265344 I see now that you want real solutions. My answer is for complex ones and for your first version.
For the second one use Sys:=(lhs-rhs)~(sys);  but the conclusion is the same.
For real solutions it's more difficult; don't use a1>0 because the algorithms used by Maple are different and the system is too large for them.

For real solutions the DirectSearch approach could be OK but we must be sure not to obtain just an approximation of the null solution.

@Kitonum Unfortunately we should admit that this is just a speculation, Maple does not manage such problems. It does not work if we write {Limit(u(x, t), x=infinity)=0, Limit(u(x, t), x=-infinity)=0} or  {limit(u(_X, t), _X=infinity)=0, limit(u(_X, t), _X=-infinity)=0}.

Also, it does not work for A := int(diff(sin(x)*u(x,t),x), x=-infinity..infinity, continuous);

@mmcdara Once we know the cause, a workaround is simple. In your code, instead of instance := sol(...), just use:

instance := convert(sol(parameters=data), `global`);

@Earl The use of complex numbers is not essential, it is just a convenient way to obtain orthogonal coordinates (and conformal maps).