Mehdi's animation worksheet had some formatting problems. Here is a different technique to get an animated W(x,t). In this worksheet, I use all your original variable names (indeed, I simply cut-and-pasted from your two original worksheets).

Download animated_beam.mw

To get an analytic colution with numeric coefficients, use the option convert_to_exact= false. Using Mehdi's formulation of the system, the command is

ans:= dsolve(sys union BCs, {X(t), Y(t)}, convert_to_exact= false);

This will give a solution that is 12 screens long, so there will be significant round-off error in the numeric evaluation. Indeed, if you want to evaluate it numerically, it would probably be better to use Mehdi's dsolve(..., numeric) solution. If you do evaluate the analytic solution at specific real values of t, you will get small imaginary parts. These are due to round-off error, and they should be ignored or discarded. Like this:

evalf(eval(ans, t= 1));

simplify(fnormal(%), zero);

You have five boundary conditions, but you should have only four.

It can be done like this, for example. This code does the same thing as the code that you posted above.

restart:
N := 10; #number of oscillators
K := 100; #number of times process is run
phi := 5; #bump given to oscillators when one fires
alpha := .5; #constant that affects extent to which size of firing populations affects bump
R:= rand(0..100); #100 should be a named constant
S := Array([seq(Record[packed]('val'= R(), 'sensitivity'= `?`, 'strength'= `?`), j= 1..N)]);

to K do
     t:= 100 - max(seq(S[j][val], j= 1..N));

     mu:= 0;
     for j from 1 to N do
          S[j][val]:= S[j][val] + t + phi;
          if S[j][val] > 99 then S[j][val]:= 0 end if;
          if S[j][val] = 0 then mu:= mu+1 end if
     end do;

     for j from 1 to N do
          if S[j]<>0 then
               S[j][val]:= S[j][val] + exp(alpha*mu);
               if S[j][val] > 99 then S[j][val]:= 0 end if
          end if
     end do
end do;

When you want to add a finite specific number of terms (rather than perform a symbolic summation) you should use add instead of sum. Then you will get the correct answer. By using sum, you are causing the premature evaluation of 0^(2*k), which evaluates to zero,  k being initially unknown. With add, the terms are not evaluated until the values of k are substituted, and then 0^0 evaluates to 1.

This issue has been discussed on Maple forums for decades.

Like your factor issue of today, there is also a workaround for this problem using frontend. This time we do freeze I, which frontend does by default. To freeze it means to treat it as just a name. In the factor issue, we froze the Pi (frontend froze it by default) but not the I.

frontend(convert, [[x, [I*x-1, 1]], parfrac, x], [{`+`,`*`,list}]);

Here is how to work around this issue with frontend:

frontend(factor, [Pi*(x^2+1), {I}], [{`+`, `*`}, {I}]);

I can't come up with logical argument about why things are the way they are with factor, except to say that perhaps it was too difficult for it to be designed otherwise, and that the designers knew that these idiosyncracies could be circumvented with frontend.

f5:= (a,b,c,d,e)-> a^2+b^2+c^2+d^2+e^2:
f3:= (ab, cd, e)-> ab[1]^2 + ab[2]^2 + cd[1]^2 + cd[2]^2 + e^2:
f5(1,2,3,4,5);
                                                   55
f3([1,2], [3,4], 5);
                                      55

There is a Maple procedure to automate this: codegen:-packparams:

codegen:-packparams(f5, [a,b,c], ABC);

My method, similar to Acer's, takes 23 seconds. Note that I selected a different NAG integration routine, appropriate for infinite intervals, and set the upper limit to infinity (because I assumed that that was your original intention). Note the capital-I Int. That's the key to getting numeric integration.

with(plots):
z:= 2*Int((sin(2*y)-sin(y))*cos(y*x)*exp(-y^2*t)/y, y = 0 .. infinity, method= _d01amc)/Pi;
animate(plot, [z, x = 0 .. 10, y= -.1..2, gridlines= false], t = 0 .. 1, frames = 100);

I see numerous syntax errors, and I don't know why you can't find them yourself at this point.

1. There should be a semicolon, not a comma, after local bb
2. i also should be declared local
3. There should be no = after i in the for clause
4. You need a space before end if
5. You have an extra colon in y::= x[i]^2

Fix those, then I'll help you with the rest.

You can't use the word error as a name in Maple code. (Well, you can't do it directly at least.) Use err instead.

You're making this unnecessarily complicated. Just use dsolve and plot the function that it gives you for each value of b:

restart:
ODE:= b-> 2*diff(y(t),t$2) + b*diff(y(t),t) + 9*y(t) = 0:
ICs:= y(0)=0, D(y)(0) = -3:
Y:= b-> rhs(dsolve({ODE(b), ICs})):
b||(1..3):= 1, sqrt(72), 9:
plot(['Y(b||k)' $ k= 1..3], t= 0..2*Pi, legend= ['b = b||k' $ k= 1..3]);

Your syntax for a Matrix constructor in HomRot is completely wrong. It should be

HomRot:= theta->
     < < cos(theta)  | sin(theta) | 0 >,
       < -sin(theta)  | cos(theta) | 0 >,
       < 0               | 0              | 1 >
     >;

There are a few other syntaxes available also. Without seeing your code for Trans and HomSquare, I can't help you further.

Do you intend for the function to be zero outside the ranges that you defined in the piecewise? One thing that you can do is simply add a plot range:

plot(piecewise(0 <= t and t < 2, 2-t, 2 <= t and t <= 3, 2*t-4), t= 0..6, axes= boxed);

You can select individual curves in a plot with the mouse. Then right click for the context menu. Then select the Line submenu.

Programmatically, you can do something like this:

plot([x, x^2], x= 0..2, linestyle= [solid, dash]);