If eqs is a system of rational-function equations, as is the case here, then the simple command (numer@(lhs-rhs))~(eqs) will reduce it to an equivalent polynomial system. That alone is likely to give a boost to most solvers. If the system is equivalent to a linear system, as is the case here, then it's very likely (although not 100% guaranteed) that those polynomials will be linear. This will almost certainly improve the performance of any solver. Thus:

eqs:= {(T[1]-T[0])/(10000-T[1]+T[0]) = -2.000000000,
        (T[2]-T[0])/(20000-T[2]+T[0]) = 0,
        (T[3]-T[0])/(50000-T[3]+T[0]) = 50,
        .1*T[0]+.3*T[1]+.55*T[2]+0.5e-1*T[3]-.5 = 0}:
Sol:= fsolve((numer@(lhs-rhs))~(eqs));
        Sol := {T[0] = -8450.48039, T[1] = 11549.51961, T[2] = -8450.48039, T[3] = 40569.12746}
To be rigorous, we must verify that the solution works in the original equations because there is a
possibility of a 0/0-type undefined, which'd be flagged with a divide-by-zero error during the 
verification.
eval(eqs, Sol);
      {-2.000000000 = -2.000000000, 0. = 0, 0.000003 = 0, 50.00000036 = 50}

@Rariusz You need to translate the phrases "time", "input signal", "position of rotor", and "speed of rotor" into the variable names that you've used in the code. My guesses would be time = t, input signal = V, position of rotor = theta. I don't know about "speed of rotor"; perhaps that is diff(theta(t), t).

If V is a function of t, then V can't be used in a parameters option to dsolve. The syntax will allow it, but the results will be meaningless. You need to construct a function to interpolate V. This function can be symbolic or purely numeric. If it's purely numeric, than you'll need to declare it with dsolve's known option.

Your parameters should be [J, b, k, R, L]. These variables should not be assigned values prior to the dsolve call. By using Statistics:-Fit with procedure-form input and using the given values as initial values for these five variables, you can fit the parameters to the data in your file.

Unlike its ancestral language Algol, Maple's syntax doesn't allow a for-do statement in place of an expression sequence. Instead, use the seq command to create expression sequences inline. Replace your

for a in 'solve(diff(func, x) = slp, x)' do 'MMLV1'='MathML[Export]('y'=Tangent(func,x=a))' end do

with

seq('MMLV1'='MathML[Export]('y'=Tangent(func,x=a))', a= solve(diff(func, x) = slp, x))

You'll also need to adjust the unevaluation quotes, but I don't know enough about Maplets to fully correct them.

There are many methods to do this. I think that the one that is closest to what you were thinking is

map(eval~, [a*x, b*x], x =~ [p, q, t]);
 

While it is totally true, as Acer said, that do is not part of an if statement and that you don't want a loop in that position, that is not the proximate cause of your error. The cause is that every do needs to be matched with an end do (or od, or until (in Maple 2018)). If you replaced do return a with do return a end do, then your procedure would work, although there is no good reason to do this, and it makes your code confusing to read.

Note that your entire procedure (without the remember table) could be replaced by the one-liner: 

GCD:= (a::integer, b::integer)-> `if`(b=0, a, thisproc(b, a mod b)): 

The degree of a polynomial with  respect to x is analogous to its base-x logarithm. The logarithm of 0 is -infinity. Also note the exact correspondence between the identity stated by VV and log(f*g) = log(f) + log(g).

Here's a procedure that does all the constants in an expression at once, representing the result in the current system of units:

`combine/Constant`:= e-> 
   combine(
      evalf(
         subsindets(
            e,
            And(name, satisfies(x-> ScientificConstants:-HasConstant(x))),
            ScientificConstants:-Constant, 'units'
         )
      ),
      'units'
   )
:

Then, to use it on an expression such as your Rb, do

combine(Rb, 'Constant');

I realize that this is not an ideal solution; there ought to be an easier way, but I don't think that there is.

As far as I can tell, the convention is that when any Maple command returns a global name whose form has not been specified by the user, that name begins with an underscore _. The example that you show does follow that convention. If you find any command that deviates from that convention, it would be worth posting about it. The names beginning with underscore in any expression e can be extracted by

indets(e, suffixed(_));

What makes you think that your command  didn't work? The number returned by Export is the number of bytes written to the file.

Here's another way:

restart:
T:= table(sparse):
#Recast the permutation vector as a 3x3 matrix:
pm:= ArrayTools:-Alias(output((P:= Iterator:-Permute(1..9))), 0, [1..3, 1..3]):

#Then conversion to base-10 and summing can be done as a single vector-matrix-vector 
#product:
(o,t):= (<1|1|1>, <100, 10, 1>):
in P do T[(v:= o.pm.t)]:= T[v]+1 od:

#Group each number of occurences by the sums that attain it:
sort(
   [indices((lhs~)~(ListTools:-Classify(rhs, [indices(T, 'pairs')])), 'pairs')], 
   'key'= -lhs
);

#This is useful for further analysis:
X:= Statistics:-RandomVariable(
   EmpiricalDistribution(
      [indices(T, 'nolist')], 'probabilities'= [entries(T, 'nolist')]/~9!
   )
):

Statistics:-DensityPlot(X);

This shows both an early stage of convergence via the Central Limit Theorem and an amazing fractal pattern.

Here's one way to do it:

restart:
macro(DT= DocumentTools):
DT:-Do(%TextArea2 = eval(DT:-Do(%MathContainer0), [x= %TextArea0, y= %TextArea1])):

I am not very familiar with DocumentTools, and I suspect that there are numerous simpler ways to do this.

Of course there is no algorithm for that. That's light-years beyond the capability of any human or computer mathematician.

You should use the parameters. One reason is that if there's a bug in your code that's caused by misspelling a parameter, it can be easily found with maplemint. Why quibble about typing a hundred words, which is like a medium-length paragraph on MaplePrimes? This one has about fifty.

If I were writing the code, I'd put a comment, just a few words, after each parameter. The code will behave better if you respect it rather than mistreat it. Respect your code!

Every example statement that you wrote is correct except for this one:

option if x1 = .. else... end; 

option is not supposed to be followed by a statement. It is supposed to be followed by a keyword. You can find the keywords that have predefined meanings at ?option. You can also use your own keywords, which you can give meanings, or simply use for documentation. For example, I often do

option `Author: Carl Love <carl.j.love@gmail.com> 7-Dec-2018`;

The difference between doing this and using a comment is that the options are integral and easily accessible parts of the "compiled" code.

You asked about module() end module. While this construction is syntactically valid, and it does return a valid module, I can't immediately think of a practical use for it. But things 0, {}, [], [][], (), and NULL are all quite useful, so I suspect that the empty module can be useful also.

You want to iterate an operation over a container of operands. There are many ways to do it. Two ways are

Rank~([G1]);

or

map(Rank, [G1]);

Your original command returned G1 as a sequence of matrices. Putting G1 in square brackets, [G1], makes it a list. Sequences are the most unwieldy of containers, and they can't be used for most forms of argument passing.