@asa12  The blue particle is at the origin. The red particle rotates around the blue one. x0, y0 are the initial coordinates of the red particle:

RotationIn2D:=proc(x0, y0)
local A;
uses plots;
A:=t-><cos(t),-sin(t); sin(t),cos(t)>;
animate(plot, [[[convert(A(s).<x0,y0>,list)[],s=0..t], [convert(A(t).<x0,y0>,list)]], style=[line, point], linestyle=3, color=red, symbol=solidcircle, symbolsize=17], t=0..2*Pi, frames=90,background=plot([[0,0]], style=point, symbol=solidcircle, color=blue, symbolsize=17), axes=none, scaling=constrained); 
end proc:

Example of use:

RotationIn2D(2,1);

Edit.

@Markiyan Hirnyk  Thank you for attention! The reason for the error is that there are just no solutions for these parameters. I made the necessary adjustments to the procedure code. Now it returns an empty list in such cases.

For example 
HeronianTriangles(170, 165);

returns the solutions.

@acer Thanks. With Maple it can be made easier

restart;
sum(cos(2*Pi*k/n), k=0..n-1);
                                                                          0

But even easier to prove this identity manually, if we note that the sum of  n unit vectors with the beginning in the origin and whose ends lie at the vertices of the regular n-gon is equal to  0  (this identity is simply the projection of this vector equality on the horizontal axis.).

@ThU  Similar identities are often much easier to prove manually than with Maple. Here is an example in which the result is also obviously 0 :

simplify(sum(cos(2*Pi*k/214748364), k=0..214748363));
     Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

simplify(add(cos(2*Pi*k/214748364), k=0..214748363));
     Error, Maple was unable to allocate enough memory to complete this computation.  Please see ?alloc
 

@acer    @vv  Thanks for the helpful comments.

 @vv    `GAMMA/magic`:=100,100,infinity,50;   # I did not understand what that meant. Is it documented somewhere?

@mmcdara  Your procedure is contrary to the condition. According to the condition
if x<M_i then M_i*2
else M_i  
those elements of the list must be doubled, which is more than a given number  x . In your example  x=5

@Markiyan Hirnyk  We all know that the analytical skills of Mathematica in a number of areas are unsurpassed. This is not news. So once again, do not spoil the mood of all Maple fans.

Happy New Year!

What is a[1] , a[3]  and  p[3]  in your diff. equation?

@Markiyan Hirnyk 

I saw your comment after I sent my answer and should note the following:

1. Your answer contains a significant error: for  t-variable the period is equal to  8*Pi  and not  2*Pi.

2. I would not rely on Optimization:-Minimize command, because it reliably finds only local extremes, not global extremes.

3. I consider the most psychologically convincing argument (at least for me personally) a neatly constructed plot. There is almost nothing in your comment regarding graphics.

 

@Christopher2222  Unfortunately, in another example, less obvious, the procedure does not work correctly.

Digits:=20:
L:=[[116,1], [139,1], [189,1], [208,1], [282,1], [471,1], [724,1], [782,1], [885,1]]:   
m:=add(L[i,1], i=1..nops(L))/3;  
evalf[5](m);   
BinPacking1D(m, L);

                                                        3796/3
                                                        1265.3
              [[208] = 1, [282, 885] = 1, [471, 724] = 1, [116, 139, 189, 782] = 1] 

 
The correct answer is   [[782, 471], [189, 885, 208], [724, 282, 139, 116]]

The sums of the numbers in sub-lists   [1253, 1282, 1261]

@n11n11  See the discussion of a similar example in this thread

1. You must submit all of this not as an image, but as a text that can be copied to the Maple window and then work with it. I do not think there are people who want to reprint it all.

2. You wrote "how i can introduce this function?". What function do you have in mind?
And how to understand this  "m is power and not derivative, for example m=1 is derivative of order 1." ?

@n11n11  The reason is that Maple 2015 does not have a built-in tool for solving elliptic equations with boundary conditions. Therefore, you must either use the newer versions of Maple, or you can solve it partially manually, using known methods. 

@Rouben Rostamian  I just narrowed the range for y-variable to  0..Pi  in the plot to better see one arch of a sine wave.

@Jjjones98 

restart;

y:= x-> ((1 + a*x + b*x^2)/(1 + c*x + d*x^2))*(ln(sinh(x)^2 + cosh(x)^2)):

Sys:={seq(coeff(series(y(x), x=0, 7), x^n)=0, n=3..6)}:

Sol:=solve(Sys);

series(eval(y(x),Sol), x=0, 7);

P:=convert(%, polynom);

eval(y(x), Sol)=P;

solve(%, ln(sinh(x)^2 + cosh(x)^2));