It would be very difficult to get your expression into exactly the form that you want. But here are some easy tricks that get it fairly close.

Download ln_of_product.mw

See ?readdata . It's like ImportMatrix, but it lets you specify a separate datatype for each column.

To get the complex roots of a polynomial p(z), use fsolve(p(z), complex). The Maple command for the argument is simply argument. The modulus is the same as the absolute value, which is abs. The maximum value in a list can be found with max.

Below, I also included a plot of the roots of the two polynomials in the complex plane.

Download arg_and_modulus.mw

Just make the existing code into a procedure with parameters mu, rho, and g. Like this:

restart: Digits:=20:

ThreePlots:= proc(mu, rho, g)
local
     C_0:= 0.4, d:= 0.0002,
     rho_v:= 1000.0, m:= evalf((4/3)*Pi*rho_v*(d/2)^3),   

     F_D:=proc(v::list)                 
     local
          vx:= v[1], vy:= v[2],
          k:= -evalf((3*Pi*mu*d+(Pi/8)*C_0*rho*(d^2)*sqrt(vx^2+vy^2))/m)
     ;      
          [k*vx,k*vy]
     end proc,                                                      

     Eskritt:=proc(vo::list,h)  
     local F:= F_D(vo);
          [vo[1]+h*F[1], vo[2]+h*(F[2]-g)]
     end proc,

     v_0:= 1.0, h:= 0.002,
 
     OnePlot:= proc(phi, N, C)
     local
          v0:=evalf([v_0*cos(phi), v_0*sin(phi)]),
          i, j, v, r
     ;
          v[0]:=v0;
          for i to N do  v[i]:= Eskritt(v[i-1],h)  end do:
          r[0]:=[0,0]:
          for i to N do  r[i]:= r[i-1]+h*v[i-1]  end do:
          plot([[r[j] $j=0..N]], color= C)
      end proc,

      p1:= OnePlot(Pi/4, 65, green),
      p2:= OnePlot(Pi/5, 55, red),
      p3:= OnePlot(Pi/6, 50, blue)
 ;     
     plots:-display(
          [p1,p2,p3], labels=["",""], scaling= constrained
     )
end proc:

ThreePlots(1.8*10^(-5), 1.2, 9.81);

Then call it again with the new values of mu, rho, and g.

You need to specify a value of alpha in the same places as you specified M. I used alpha=1, but the resulting plots were not within your view. You should eliminate the view option until the plot is otherwise as you want.

b:= A[..,3]:
C:= A[..,[1,2,4..11]]:
x:= LinearAlgebra:-LinearSolve(C,b):
x[8];

You need to define the function f like this:

f:= (t,v)-> 0.0207*v^2 - 893.58;

and not like this:

f(t,v):= ....

Perhaps you are confused by the ?piecewise notation in which the integral answer is presented. You can convert it back to abs form, and you can also convert it to ?Heaviside form.

Download convert_abs.mw

The Interp command that I showed for one of your previous questions works just as well in the two-variable case. So here's your problem (iii):

restart:
U:= [0,1,2]:
V:= [x^2+7, x^3+2*x+3, x^3+5]:
Interp(U,V,y) mod 11;

To see the algorithm for doing this, enter

showstat(`mod/Interp/Interp`);

You just need to pay attention to lines 11-26.

G:= GF(7, 3, x^3+x+1):
G:-`/`(G:-ConvertIn(x^2));

Your problem 1 can be solved by a single Maple command:

Interp([0,1,6], [1,5,2], x) mod 7;

It's entered just like your problem from earlier today.

PDE:= diff(r*diff(T(r,t),r),r)/r = diff(T(r,t),t)/beta:
pdsolve(%, build);

Here's a solution that makes use of parallel processors.

PowerSet:= proc(S::set)
local P,x,s;
     P:= {};
     for x in S do  P:= P, Threads:-Seq({s[], x}, s= {P})  end do;
     {P}
end proc;

pdsolve(diff(y(x,t),x$2)+diff(y(x,t),x)/x=diff(y(x,t),t)/alpha);

Here is why your subs did not work. Consider the expression x+2+sin(x+2) as a tree (in the mathematical sense of tree). The root is the whole expression. The first level of branching divides into three nodes: x, 2, and sin(x+2). The x and the 2 are terminal (leaf) nodes. The sin(x+2) splits into two nodes: the sin and the x+2. The sin is a leaf, and the x+2 splits into two leaf nodes. (Pause now to sketch the tree.) The expression which gets substituted for, i.e., the left side of the = in the subs command, has to be a node in that tree. The x+2 only appears as a single node when it is the argument of sin. The other x and 2 get split immediately from the root.

Does that makes sense?