The obvious works:

eqn:= diff(x(t), t) = 0:
subs(diff(x(t), t)= v(t), eqn);

Your fprintf statement has four %gs, but you only have three numeric values (three evalfs).

ex:= x/2+y/2:
content(ex)*``(primpart(ex));

To extract the leading square submatrix of order M, use A[1..M, 1..M].

Any of the following:

subsindets(y=a+b, atomic, f);
subsindets(y=a+b, name, f);
evalindets(y=a+b, atomic, f);
evalindets(y=a+b, name, f);

See command ?fnormal .

See ?numapprox,minimax .

If you don't mind extra parentheses in the display, you can do this

``(3) + ``(7) + ``(1);

If you want it without the parentheses, then it is significantly more complicated, but it can be done with the Typesetting package.

Your equation is fifth order in alpha. There is provably no possible general solution for fifth-order and higher-order polynomials. Since you have parameters, there is no numerical solution either.

When Maple makes up a variable name, it begins that name with a underscore to help distinguish it from user-supplied names. _Z takes the place of the solved-for variable in a RootOf expression. That is, _Z takes the place of alpha. A RootOf is Maple's placeholder for the root of an equation that Maple in unable to solve.

First, you need to change your syntax. To apply sqrt to two arguments, the closest syntax to what you had is

sqrt~([c^2, r^2]);

The reason that Maple does not simplify these square roots is that it does not know the sign of c and r or even that they are real. You can supply this information with an assuming clause:

sqrt~([c^2, r^2]) assuming c > 0, r > 0;

or

sqrt~([c^2, r^2]) assuming positive;

There is also a symbolic argument that can be give to sqrt:

sqrt~([c^2,r^2], symbolic);

add(a[i]*mul(`if`(j=i, 1, s-b[j]), j= 1..n), i= 1..n);

To obtain better understanding of what minimize does, enter your expression in exact form:

f:= x*sin(4*x)+11/10*sin(2*y):minimize(f, x= 0..10, y= -10..10, location);

Note that the returned answers are exact. I think that it finds the zeros of the gradient and evaluates the function at them to find the minimum evaluation. It does not use a numerical method.

Answering your second question, Explore can be used with piecewise just as with any other mathematical expression. For example,

Explore(plot(piecewise(x < a, -1, 1), x= -10..10, discont), parameters= [a = -10..10]);

I don't know what's going with your first question. It always appears in the same document for me. Maybe there is a default setting for it.

The command with does not work correctly inside procedures. Eliminate it, and replace the jacobi line with

numtheory:-jacobi(A,B)

Indeed, the entire procedure is superfluous, and could be replaced by the line

jacob:= numtheory:-jacobi;

First, remove Pr from your list of parameters in the second line. Second, change the line where you define PDE to

PDE:= unapply({pde||(1..3)}, Pr);

Then replace your last two lines with

PrList:= [0.71, 7, 10]:
Colours:= table(PrList =~ ["blue", "red", "green"]):
for Pr in PrList do
     pds:= pdsolve(PDE(Pr), IBC, numeric);
     Plots[Pr]:= pds:-plot[display](
          u(y,t), t=0.5, linestyle= "solid", colour= Colours[Pr],
          legend= sprintf("Pr=%2.2f", Pr),
          title= "Velocity Profile", labels= ["y", "theta"]
     )
end do:
plots:-display([seq(Plots[Pr], Pr in PrList)]);