solve has problems with so many variables and almost crashes Maple (and Windows).
Replace solve with LinearSolve this way:

V:=[indets(Eqns1)[]]:
AB:=GenerateMatrix(Eqns1,V,augmented):
Vsol:=LinearSolve(AB):
V=~Vsol;

The simplest way seems to be:

impdiff := (f,y,x) -> -diff(f,x)/diff(f,y):
impdiff( (lhs-rhs)(eqn), x, y);

If you simplify the result the expansion appears again. But in Acer's solution too.

In your procedure EEA you cannot use for example a*b.
You must use ED[`*`](a, b)

KroneckerDelta   is a tensor supposed to be used in Physics, being related with the spacetime dimension.
Use piecewise(m=0,1)  instead; it is even shorter!

The unpleasant fact is that the integral is difficult to manage with IntegrationTools:-Parts because it is transformed in terms of WhittakerM.
We must do part of the work by hand to obtain the final result 

(1 - (b+1)^(-a))*GAMMA(a)/b.

For programming the solution is to use neutral operators.

`&x` :=LinearAlgebra:-KroneckerProduct;
A := <a,b;c,d>;
B := <1,2;3,4>;
A &x B &x A;

It works if you correct  2x  to  2*x.

Let f be a superposition of two functions f1, f2 . i.e. f(x) = g(f1(x),f2(x)) .

If f1 has period T1 and f2 has period T2 then f will have generally as period the lcm of T1 and T2, provided that T1, T2 are commensurable (i.e. T1/T2 is rational).
In your example we must know Omega. E.g. for Omega=1 the function is not periodic.

CommonPeriod:=(T1::Not(0),T2::Not(0)) -> `if`(type(T2/T1,rational), abs(numer(T2/T1)*T1), infinity);

For example, the function    cos(4*x) + sin((3/2)*x)     will have the period
CommonPeriod( 2*Pi / 4,  2*Pi / (3/2));
      4*Pi

A convert and a correct assumption are needed.

simplify(convert(exp(I*t)^q, exp)) assuming t > -Pi, t <= Pi, q > 0, q < 1;

Why don't you use the monomial order lexdeg([x,y,z], [u,v,w]) ; it is used exactly to eliminate x,y,z.

Your monomial order can be obtained:

t2:=plex(u,v,w);  t1:=wdeg([1,1,1,0,0,0],[x,y,z,u,v,w]);

t:=prod(t1, t2);  # your order

sat:=a -> satisfies(u -> (op([0,0,0],u)=D and op([0,1],u)=a)):
difford:= (e,a) -> max(nops~(map2([op],[0,0,..],indets(e,sat(a))))):

# Example
expr:=(diff(a(x, y), x, y))*x*y+(diff(a(x, y), x, x, y))*x+diff(b(x, y), x, x)-sin(z):
expr:=convert(expr,D):
has(expr,a); # depends on a?
                              true
has(expr,c); # depends on c?
                             false
difford(expr,a);
                               3
difford(expr,b);
                               2

For example, the principal branch with ~ 3 digits accuracy, for x>0:

W = 1.45869*ln(1.2*x/(ln(2.4*x/(ln(1+2.4*x))))) - 0.45869*ln(2*x/ln(1+2*x))

You probably know that  f(n) = O(g(n))  means  limsup( f(n)/g(n), n=infinity) < infinity

provided that f,g>0  and  n -->oo.

Now, Maple does not have limsup; we shall use limit instead (in most cases it's OK).
So, your "equation" is equivalent to

limit( (n^2/2 + 2*n - 1)/n^2, n=infinity ) < infinity;
        1/2 < oo
It is true. What do you want to solve here? Do you mean prove?

LinearAlgebra works fine. You must know that it uses Matrix and Vector (capital M and V) instead of the old matrix and vector.
I suspect that you have used matrix. If this is not the case, please post your example.

The equation is equivalent to

x ln(x) = ln(4) + 2n Pi I <==>  ln(x) exp(ln(x)) = ln(4) + 2n Pi I <==>

ln(x) = W(m, ln(4) + 2n Pi I)   and  Im( W(m, ln(4) + 2n Pi I) )  in (-Pi , Pi].

Here m,n are integers.

But  Im( W(m, ln(4) + 2n Pi I) )  in (-Pi , Pi]   iff  m=0  (for n in Z).

Note that m is an arbitrary integer if all the branches of the logarithm (or equivalently of x^x) are considered.