map[3](op@FileTools:-ListDirectory,currentdir(),'all','select'=~["*.mw","*.eps"]);

You odes and initial conditions need to be together in a set (or list). You also have three functions A(t), Ab(t), R(t) and only two odes; I'm assuming Ab(t) was meant to be A(t). Then it works. (I made c3 smaller to see something on the plot.)

dsolve_problem.mw

Your error disappears if you run the worksheet after a restart. Your graphs are all the same because although you change S[t] when you define your substitutions p1, p2 etc., there is no S[t] in the expression you are substituting into (which I've renamed ff since you are using f(x) for other purposes in your worksheet). You should be careful about using both S[t] and S as variables; if you set a value for S then S[t] will not be defined -- to solve this use S__t instead of S[t]; they look almost the same on output (the first has the t in italics).

To change the view for the plot, use view=[0..1,0..1].

sstf.mw

After transforming to xi, evalc(Re()) and evalc(Im()) work as you would expect, and you get Eq. (1.5). (I find the pdes easier to manipulate if you leave off the =0, which is usually assumed anyway for many Maple commands, such as pdsolve.)

Download PD_OD2.mw

You can subtract a common term from each side of an equation or inequality using elementwise subtraction, assuming you know what term you want to subtract. If you don't, it is simpler to subtract it all, which is essentially @nm's answer. (I would typically set up an equation or inequality in the ineq3 form originally, then you do not have this issue.) Here it turns out the simplification makes the inequality harder to solve. 

For the assumptions you gave in the worksheet, solve gives two cases.

Download Q_12.mw

The other question about phase shifts has now been deleted. .pdf of it is here: phase_shift.pdf

Download phase_shift.mw

To get Maple to work this out, turns out to be surprisingly difficult, if you want to avoid inputting things like the cosine rule that you expect Maple to know. Here's a solution. Really, it should be easier.

Download polygon.mw

If I trust isolve (?), then there is only one solution, with the tangents 1, 2 and 3. In the figure below tan(A) = 3, tan(B) = 2, tan(C) = 1.

Download tangents.mw

The exponentials are gone in eq22, so it has worked. I assume that after the substitution, the expression with V simplifies so that there is no longer a V.

If fact you can verify this by doing simplify on eq2.

p.s. use indets(eq22,name) to only see the variable names and not more complicated expressions.

@salim-barzani This iterates over all possibilities of mu[1]=0,1;mu[2]=0,1; etc, and is the only way I can make sense of it.

Download formula.mw

In Maple, u[1/2](t) and u[0.5](t) are not the same so you cannot mix them. I changed all the 0.5's to 1/2 and it seems to work now.

error.mw

It's a bug. A workaround is to use

ans := SolveTools:-PolynomialSystem(eqs,
  {p, q, a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], k[1], k[2], k[3], m[1], m[2], m[3]}, explicit)

If I understand correctly, you want to solve the partial differential equation diff(u(x,y,t),t)-diff(u(x,y,t),x)=z. The answer depends on an arbitrary function f__1 at the end. So if you have some boundary or initial conditions you can refine the solution.

Download pdsolve.mw

Here is one way. pointplot has options to control the shape and size of the point. display is used for combining two or more plots. Notice the with(plots) at the beginning. I'm not sure what you mean by "plotting the distance".

Download How-to-plot-distance-midpoint.mw

For numerical use (not necessarily for plotting), plottools:-rotate rotates an nx3 Array (datatype=float[8]) of points if you enclose it in a CURVES or POLYGONS call. I'm assuming the alpha, beta, gamma are the Euler angles, or closely related. (I've only actually used the other form of plottools:-rotate where you specify 2 points on the rotation axis and one angle.)

Download Euler.mw