@nm PDEtools:-Solve says clearly that the multiplicity is ignored.

@MapleEnthusiast In E is computed the regular inverse (and the result is simplified). The matrix C-A is invertible, so the pseudo inverse equals the regular inverse and using method=pseudo just complicates the computations.

@MapleEnthusiast Your system is a simple polynomial system and homogeneous linear  in the three unknowns. The rank being 2 (in general) it has a unique solution (modulo a multiplicative constant). Obviously It cannot be solved in another way.

So, your new questions make no sense.

P.S. You have very long names in your system. Not knowing anyway their meaning, it would have been much easier for us to see x1,x2,x3 (unknowns), a1,a2...,an (parameters). 

@zphaze  It's about maths, not other CAS-es (check again with Mathematica and Maxima!).
The equation (in z)   arctan(z) = a   has a real solution iff -Pi/2 < a < Pi/2. What is a in your case?

@RohanKarthik I would  have formulated the question this way:
Among all the terms A*x^a*y^b  satisfying a - b \in {0,-1}  of a polynomial f(x,y) ,
find the one for which a+b is minimal.

Or, solve:

min { a+b  |  x^a*y^b is a monomial in f(x,y) and a - b \in {0,-1} }.

@zphaze It is not possible to guess user's intentions so easily. f and f(x) may coexist in a legit expression:

plot(sin + sin(1), 0..2*Pi);

is equivalent to

plot(sin(x)+sin(1), x=0..2*Pi);

@ecterrab It is a bit strange to have 2 kinds of inert versions without being equivalent.
E.g. %int is transformed into Int by IntegrationTools but %gcd does not work under mod.
It should be chosen a wise and uniform decision about this. Of course, Gcd etc could be seen as "special" inert forms, and that's it!
 

@mthkvv 

restart;
p := 10^140:
a := modp1(ConvertIn((2*10^139+13)+x^1000-x^2,x),p):
save(p, a, "d:/tmp/pab.m");
restart;
read "d:/tmp/pab.m"; # ok
modp1(ConvertOut(a),p)[1] - (2*10^139+13); # 0
 

@acer Even after local gamma,  evalf(gamma(0)); ==> 0.57...   etc
due to `evalf/gamma`;
 

@janhardo You should be aware that Maple can help (in Calculus here) only at a formal level. To understand things you must study carefully a good textbook and be able to solve at least the simple exercises by hand. Not doing so, you are wasting your time.

@janhardo Is this a question, whether f(x) and f(y) are two functions?
You have only one f, in two variables. y is a variable, no y(x) here! 
Try to use proper terms and notations, otherwise everything becomes a mess.

@mthkvv I'd consider writing a procedure P(m) using some generated polynomials (preferably with known results).
Then call P(m) for an increasing sequence of m's inside CodeTools:-Usage and try to figure out the amount of needed memory as a function of m. [It probably also depends of how dense the polynomials are].

@Pascal4QM `--` is an assignment operator in Maple 2020 (2019?)

I don't think that a teacher would be very happy to see one of his students writing: -3 - -5  (unless he missed the class about parantheses).
The next level would be:  --5+-7-+2++3*-1
 

@mmcdara Unfortunately it is easy to find modern "applied" math formulations: "The function y = f(x) ..." or "Let f(x) be the function ...".
Even Maple is not very careful with the math terminology. E.g. a better name for the type function would have been funcall.