@mostafaorooji It can be done with coeffs like this:

[coeffs(rhs(ans), indets(ans, suffixed('_C', integer)), 'C')];

This works regardless of the presence of any specific function such as exp. The returned order will correspond to the integers affixed to _C, (which are not necessarily consecutive). If there's any doubt about the order, it will be explicitly listed in C.

A Maple module is a piece of high-level Maple code, essentially a type of procedure. Modules are not stored as machine-language code, and are not stored in .dll files. Some modules may have a .dll file associated with them, but I didn't want you to think that there was an inherent connection between the two.

@vv Huh. I wonder if that's a change made for Maple 2018.2. I still have 2018.1, and RealRange(-infinity, infinity) returns unevaluated. Anyway, I changed the code to accomodate this situation, so please test it.

@vv Please apply lprint to the results containing real, and tell me what you get.

@Harry Garst It would not have occurred to me to use this method as a starting point for factoring.

@adel-00 It's just a trivial change:

plot([rhs(f)^2, Y, Y= 0..10], labels= [typeset(epsilon^2/`4`/Pi), Y]);

@vv I corrected the code to accomodate your ex1, and the new code is in my previous Reply. Regarding your ex2, the result that I get is RealRange(-infinity, 1). Unless I'm having a huge mental lapse, that's the correct answer. Is that what you get?

@acer I did both things, and here's the final product:

SimplifyIntervals:= module()
description
      "Simplifies arbitrarily nested set expressions of real intervals expressed with RealRange"
      " and/or finite discrete sets. Returns a union of disjoint intervals and discrete points."
;
option `Author: Carl Love <carl.j.love@gmail.com> 26-Nov-2018`;
local
       x::identical(x), #bound variable for `solve`
       
       #1. Declare a type specific to this module.  2. Hook into `simplify`.  
       ModuleLoad:= proc($)
          TypeTools:-AddType( #recursive type: put base cases 1st!
                 'RealIntervals', 
                 Or(  
                    specfunc(RealRange), set(realcons), identical(real), #base cases 
                   'specop'('RealIntervals', {`intersect`, `union`, `minus`}) #recursive cases
                )
          );
          :-`simplify/intervals`:= ()-> ModuleApply(args);
          forget(simplify)
     end proc,
     
     #Convert set expression into boolean expression with bound variable x.
     Deconstruct:= (IN::RealIntervals)->
          if IN::specfunc(RealRange) or IN=real then convert(x::IN, relation)
          elif IN::set(realcons) then Or((x =~ IN)[])
          elif IN::`minus` then And(thisproc(op(1,IN)), Not(thisproc(op(2,IN))))
          else `if`(IN::`union`, Or, And)(map(thisproc, [op(IN)])[]) 
          fi,

     ModuleApply:= proc(IN, {no_union::truefalse:= false})
     #Using 'no_union' option guarantees that output will be sorted the standard way, even if it contains 
     #discrete points.
     local P, J;
          if not IN::'RealIntervals' then return IN fi;
          #Separate 'solve's output into discrete points (P) and 'RealRange's:
          (P,J):= selectremove(type, [solve(Deconstruct(IN), x)], realcons);
          if x in J then return 'RealRange'(-infinity, infinity) fi;
          J:= sort([`{}`~(P)[], J[]], 'key'= (p-> `if`(op(1,p)::realcons, op(1,p), op([1,1],p))));
          `if`(no_union, J, `union`(J[]))      
     end proc
;
       ModuleLoad()
end module: 

Example (showing three formats for the output):

IN:= (((RealRange(-infinity, 1) intersect RealRange(Open(-1), infinity))
   minus RealRange(-1/3, 1/3)) minus {0}) union {-2,2};

simplify(IN, 'intervals');
SimplifyIntervals(IN, no_union);
lprint(%);
 

@acer Thanks. I did exactly what you did above, except that I forgot to forget. 

Thank you for at least restoring some parts of the Question.

Please do not substantially change your Questions after they have been answered. You have removed almost all the context that makes Preben's and my Answers understandable. For example, Preben refers to "parts c) and d)". How can a future reader understand that if you've removed those parts from the Question? Thus, you have reduced the value of our work. Is that fair, or ethical? Does that make sense?

Please put back the question as best as you can with your attached worksheet. If you won't, then you're not welcome to ask any more questions here as far as I'm concerned.

@Lottie 

Yes, it works for contour plotting commands. Don't be afraid to try these things yourself even if you're not  sure if they'll work.

Preben has given a detailed analysis of the stability. Here's the quick way, which can be computed automatically with Maple: For nonlinear stability at the origin it is sufficient that the function that we just plotted (dV/dt) have a local maximum at the origin. Obviously this can be seen to be true from the plot. We can verify that computationally. If you recall the multi-variate second-derivative test from Calculus III, for a local maximum, it is sufficient that its gradient be 0 and that its matrix of second partial derivatives (wrt x and y) (called the hessian) be negative definite (i.e., have all negative eigenvalues) at the origin. Thus

v:= [x,y]:
DV:= subs(v(t)=~ v, dV);
VectorCalculus:-Gradient(DV, v);
is(norm(VectorCalculus:-evalVF(%, <0,0>)) = 0);
VectorCalculus:-Hessian(DV, v= [0,0]);
LinearAlgebra:-IsDefinite(%, query= negative_definite);

I just found an amazingly well written set of "notes" on Lyapunov stability analysis. This is better than most textbooks. The author is Gunnar Soderbacka; the link is http://www.users.abo.fi/gsoderba/PhaseP/ljap13.pdf

@acer The Question originally had a link to a journal article---a link which I followed. The article was from a Springer journal dedicated to BVPs. It was dated 2012. The lead author's surname was, I think, Aylin. Amazingly enough, that crap code came verbatim from that article. These authors and their editors have no respect for the code! Obviously it wasn't vetted because it won't even execute. I found the same errors as you did. And there are numerous other issues that are not outright errors for this case, but which could become errors if one tries to adapt the code to other cases.

@student_md (OP): Please put that link back, unless some legal reason prevents you from doing so. I think that since we're seriously criticizing the article, it's considered "Fair Use" under copyright law. (Disclaimer: I have no status or right to offer actual legal opinions.)

@tomleslie I believe that my Answer does what the OP intended rather than duplicating what the OP's code actually does. He did afterall say "The conditions I tried were not working." To me, that means "The inequalities in my piecewise are incorrect."

Furthermore, I believe that he was trying to use piecewise only to do a job that is better done by min, likely because he was unaware of the existence of min.

If I got that wrong, I'll be happy to correct it.

Please clarify: Do you mean that if the return value otherwise computed by the piecewise function is greater than 10, then you want simply 10 returned?