@Joe Riel You are right.  I don't know why I thought I was getting zero.

@janhardo I suggest that you post your attempts at solving those exercises, otherwise it is not clear what level of help will be beneficial to you.

@Kitonum That's very good.  Vote up!  But do you know why that doesn't work when we replace posint with integer?

int(cos(m*t)*cos(n*t), t=-Pi..Pi, allsolutions) assuming integer;

This one returns 0 in Maple 2020.

@Thomas Richard Thanks for looking into this.

@Hamza Brhl 

Download mw.mw

@Mariusz Iwaniuk As Lopez has pointed out, Maple's numeric PDE solver handles time-dependent equations in one space dimension.  This class of problems are particularly easy to solve since the domain of the unknown function is a strip
D = { (x,t),  a < x < b,  t > 0}
in which the derivatives may be readily discretized through finite differences.

In the same way, elliptic equations in 2D are also easy to solve through finite differences provided that the domain is a rectangle.  But a rectangular domain is too limiting to be of general interest.  To be useful, an elliptic equation solver should be able to handle domains of more or less arbitrary shapes (at least in 2D).  That makes applying a finite difference discretization difficult, if not impossible.

That's why essentially all elliptic equation solvers are implemented through finite elements which is a whole different ball game.  If Maplesoft goes in that direction, it will need to add a dedicated department, and staff to implement the software and to deal with what I predict will be a never-ending stream of requests for enhancements and additional features.

@mary120 From the little information that you have provided, it appears that you are using the symbol f in two different senses: (a) as one of the unknowns in your system of differential equations, and (b) as a parameter that you have defined as 0.42. 

In the future post your complete worksheet; that's more informative than merely showing the error message.

Your unknowns, w₁, w₂, w₃, are defined on three separate intervals, as in (0,a), (a,b), and (b,c). I am afraid this is too complex for Maple's PDE solver.  In fact, before attempting that problem, consider solving the much simpler problem of a beam spanning the interval (0,2), with simple supports at 0, 1, and 2.  You already  know how to formulate the problem, but here it is for the sake of the other readers.  The indices x and t indicate derivatives with respect to x and t.

We want to solve the system of PDEs 
u_xxxx + u _{t t} = 0,    0 < x < 1,   0 < t
v_xxxx + v _{t t} = 0,    1 < x < 2,    0 < t
for the unknowns u(x,t) and v(x,t), subject to the boundary conditions
u(0,t)=0,   u_xx(0,t)=0,
u(1,t)=0,   u_x(1,t)=v_x(1,t),   u_xx(1,t)=v_xx(1,t),   v(1,t)=0,
v(2,t)=0,    v_xx(2,t)=0,
and initial conditions
u(x,0)=0,   u_t(x,0)=1,
v(x,0)=0,   v_t(x,0)=0.

This is a well-posed problem but as far as I can tell it is beyond what Maple can handle at this time.

@Earl It's good that you are looking and comparing solutions from different angles.  That's where learning takes place.

@Earl  In this worksheet I solve the problem in two different ways -- first by applying Newton's equations of motion directly, and next, by applying the conservation of energy and a formula for the centrifugal force, as you had attempted.  The results are identical but the first method is much cleaner.

The solution here is somewhat different from what I posted earlier because after carefully reading the description of the problem and your worksheets, it became necessary to replace the equation F.n=2mg  with tau(t)=2mg in my calculations.

Download roller-coaster2.mw

@Earl Your formulation looks reasonable but debugging it to find the source of the difficulty is not easy.  Just looking at the expression 

PrincipalNormal(<x(t), y(t)>, t, normalized)

which enters your equations makes my head spin.  I would rather not go there.

@Carl Love That curve does not look right as it appears that the particle gains energy as it goes along.  Perhaps the original equations were wrong, or maybe something went wrong in the subsequent calculations.  I haven't checked.

Your questions do not make sense as they are posed.  You should be more precise about what you want.

1. You have presented two equations involving x and f.  They look like G(x,f)=0 and H(x,f)=0.  The equation G(x,f)=0 implicitly defines f as a function of x.  Therefore a sensible question would be: how to plot that implicit function?
2. You may want to ask a similar question about the function H(x,f)=0.
3. You ask for "contour plots of x versus f that satisfy two following equations".  If that's what you really mean, then there is nothing to plot, because solving G(x,f)=0 and H(x,f)=0 simultaneously gives two numbers:  x=something, f=something.  (That's assuming that the system has a solution.)
4. Finally you say you want to vary the coefficient of x from 0.1 to 0.6.  However there are several occurrences of x in your equations.  Which of those coefficients you want to vary?

@nm Exporting to latex one line at a time from a large worksheet can be maddeningly exasperating.  I would prefer to convert the entire worksheet to a latex file and edit the file as needed.  The problem is, the latex code produced by Maple's export is incredibly convoluted and not easy to modify. 

@vv Thanks for the idea.  That does display the expressions in a reasonable way but I would have liked to see the result displayed as a set rather than two individual expressions. I can insert the set's opening and closing braces by editing the *.tex file, but that's a lot of work with a largish worksheet, and which needs to be repeated if something in the worksheet changes.

Another issue (which I did not include in my sample worksheet) is that the LaTeX translator ignores the setting of Typesetting:-Settings(typesetdot=true) and displays derivatives in the d/dt form.

Direct export as a PDF works better in these respects, however the monolithic PDF output is not suitable for further text processing.

Maple's LaTeX translator needs a good overhaul and I hope that someone at Maplesoft will get around to doing it.