Why is 3 any different than the 11 that I showed you how to add to every term last week?

nilaiASCII +~ kekuncirahsia;

What you want can be done with a single simplify with side relations command. First you need to freeze the expressions e^epsilon and e^(-epsilon) so that every expression in the matrix is polynomial. (Note that these aren't the same as exp(epsilon) and exp(-epsilon); however, I've continued with your erroneous usage.) The expression that you have labeled (3) I call ex. Then all that you need to do is

M1:= subs(e^epsilon= freeze(e^epsilon), e^(-epsilon)= freeze(e^(-epsilon)), M):
thaw(simplify(ex, {seq(seq(T[i]*U[j]= M1[i,j], i= 1..op([1,1], M)), j= 1..op([1,2], M))}));

Since theta(t) only appears in derivatives, it is easy to reduce the order. Change diff(theta(t), t) to dtheta(t) and diff(theta(t), t, t) to diff(dtheta(t), t):

eq1:=J*diff(dtheta(t),t)+b*dtheta(t)=K*i(t):
eq2:=L*diff(i(t),t)+R*i(t)=V(t)-K*dtheta(t):
DCMotor:=[eq1,eq2];
Sys:=DiffEquation(DCMotor,[V(t)],[dtheta(t)]);

The following command will take care of the asymptotes automatically:

Student:-Precalculus:-RationalFunctionPlot((x-1)/(x-2));

alpha__0:= x-> sqrt(d/x)/2*exp((-D__s - i)*Omega*x/V__La):
alpha__90:= x-> 3*sqrt(d/32/x)*exp((-D__s - i)*Omega*x/V__s):
alpha__theta:= (x,theta)-> alpha__0(x)*cos(theta)^2 + alpha__90(x)*sin(theta)^2:

If you intend for the i to refer to the complex unit sqrt(-1), then you'll need to change it to I.

MyArray:= (n,N)-> Array((1..N)$n, ()-> [args]);

A:= MyArray(3,3);

nilaiASCII +~ nops(nilaiASCII);

Here's how I like to solve recurrences by hand with the assistance of Maple. I start with a(n) and work backwards (towards a(0) or a(1)) with a "stunted" (i.e., partially inert) recursive procedure.

a:= n-> ''a''(n-1)/n:

Note that those quotes are two pairs of single quotes, not one pair of double quotes.

eval(a(n));

eval(%);

eval(%);

At this point, I see the pattern: a(n) = a(0)/n!

It may be most efficient to do it from the original integer representation. I haven't checked its speed, but here's a procedure to do it that way:

nOneBits:= proc(n::nonnegint)
local q:= n, ct:= 0;
     while q <> 0 do ct:= ct+irem(q,2,'q') end do;
     ct
end proc:
     
nOneBits(255);

     8

Is this what you're looking for?

OpList:= proc(ex)
local r:= [op(ex)], op0:= op(0,ex);
     `if`(r = [ex], ex, `if`([op0]=[], r, [op0, r]))
end proc:
 
Atomize:= proc(ex)
local r, newr:= ex;
     while r <> newr do
          r:= newr;
          newr:= subsindets[flat](r, Not({list,symbol}), OpList)
     end do
end proc:

Atomize(x*y+z-7^sin(w));

See also ?ToInert and ?dismantle.

They're both good answers so far, but I find this a little bit clearer. I'll suppose that your Vector system of equations is named Sys. Then do

X:= [u,v,w,alpha,beta,gamma]:
(M,Sys):= LinearAlgebra:-GenerateMatrix(convert(Sys, list), diff~(X(t), t$2)):
(C,Sys):= LinearAlgebra:-GenerateMatrix(convert(-Sys, list), diff~(X(t), t)):
(K,F):= LinearAlgebra:-GenerateMatrix(convert(-Sys, list), X(t)):

Do you mean this?

convert("0123456789", bytes);

convert~(%, binary);

map2(nprintf, "%08d", %);

Try

eval[recurse](eq, [ChgtVariables]);

You can get the leading zeros with nprintf. Here's my procedure:

myproc:= proc(p::prime, {base::And(posint, Not(identical(1))):= 10})
local n,q,r;
     if irem(base,p)=0 then return 0 end if;
     for n in numtheory:-divisors(p-1) do
          q:= iquo(base^n-1, p, 'r');
          if r=0 then return (n, nprintf("%0*d", n, q)) end if
     end do
end proc:

If you don't want the leading zeros, then return (n,q).

So, you want to print 200,000 3D plots? You simply need to change  plot3d to (print@plot3d).