simplify(expr) is simpler in the sense that the result does not contain a sqrt any more.
Using simplify(expr, size), expr remains unchanged.

p := pds:-value(output = procedurelist);
p(0, 0.1);
#        [x = 0., t = 0.1, u(x, t) = -0.587785252292473]

You may want to use

f:=(x,t)->rhs(p(x,t)[-1]):
f(0.23,0.5)
#                       -0.991900750572342

rem_d:=proc(n::integer, d::posint)
  local q:=n, q1;
  if (d=1 or n=0) then return infinity fi;
  while irem(q,d,'q1')=0 do q:=q1 od;
  q
end proc;

rem_d(294912,8)  #  9
 

f := -log[2](7/10*x)+log[3](3*x-1):
#plot(f, x=1/3 .. 40);
limit(f, x=infinity);
#                           -infinity

fsolve(f, x=0.34 .. 100, maxsols=4);
#                   0.3730125034, 16.60731835

a,b:=%;
#               a, b := 0.3730125034, 16.60731835

evalf(eval(f, x=(a+b)/2));  # >0
#                          0.339241451

answer := a<=x, x<=b
#         answer := 0.3730125034 <= x, x <= 16.60731835

Exact solution possible but only in terms of RootOfs.

Of course. The right (or usual) eigenvectors of a matrix A are given by the command Eigenvectors(A). The left eigenvectors are given by the same command but for the transposed matrix.

Actually Maple can compute the limit.

The expression of a inverse function is given in the table invfunc (and also Invfunc for multivalued inverses), see ?invfunc. When the user wants to use the inverse of one of his functions, he must define the entry in the table(s).

In Maple 2018 it works:

Rational approximations are obtained via continued fractions. They are implemented in NumberTheory:-ContinuedFraction and the older `convert/confrac`.

For example:

cf:=NumberTheory:-ContinuedFraction(Pi):
Convergent(cf, 1..5);
#    [22/7, 333/106, 355/113, 103993/33102, 104348/33215]

The actual computations are in `convert/real_rat`.

`<,>`(seq(`<|>`(seq(M[i,j],j=1..2)),i=1..2));

or

Matrix(4,4,(i,j)->M[iquo(i-1,2)+1,iquo(j-1,2)+1][irem(i-1,2)+1,irem(j-1,2)+1]);

or

Matrix(linalg[blockmatrix](2,2,[seq(M)]));

Download table-evals-vv.mw

Replace sum with add in the definition of S.

Download ysol.mw

Yes, an error or at least a warning would be better. input and output work with polynomials of any degree.

Note that when n >= p^k,  input(n) produces a polynomial (zpoly) of degree >= k.
The reduction modulo the extension polynomial is not performed.
It is however performed by the arithmetic operations in G except G:-`+`, G:-`-` 
(here G := GF(p,k)).

This is a nice example where the correct use of trigonometry is essential.
It seems to be too subtle for a CAS!

By solving the ode by hand one obtains  the solution:

Y:= piecewise(x < 0, -2*arctan((x+2)/x), x>0, -2*arctan((x+2)/x)+2*Pi, Pi)

It is easy to check that it satisfies the ODE for ANY real x.
Your solution, 2*Pi-2*arccos(x/(2*x^2+4*x+4)^(1/2)),  is valid only for x>-2.

Edit.
It can be shown that  Y = 2*arctan(x + 1) + Pi/2
(it is enough to compute int(diff(Y,x), x) + C).