@vv 

Therefore, here again we must place our trust in the axiom of choice and its equivalent theorems, as was the case in the proof of Hausdorff's completion theorem.

@Kitonum 

Simple calculation by hand:
Equation of a straight line through the given points, equation of the perpendicular bisector between the given points, zero and intersection on the y-axis give P and Q, calculating the length of the line from PQ gives 15/2*sqrt(5)

@vv 

... is of course correct. From your solution, as well as from answers to other tasks, I learned something about Maple's tools, which was also my sneaky ;-) intention. But there is a much simpler solution that does not involve a significant amount of Maple.
BTW:
I wrote an answer to your task about a proof of the theorem that the identity function is the sum of two periodic functions. Unfortunately, the generally known proof is not constructive, it is "only" an existence proof. Do you know of a concrete example?

@delvin 

I cannot see a concrete solution in your post. At best, you give a hint to a possible solution method. In this respect, a concrete solution is not given. I recommend that you supplement your procedural steps with references to well-known theorems from the theory of differential equations that you have applied. In my opinion, this would make it easier for the Maple experts here to help you with your problem. I cannot do this, as I am only a theoretician and often ask for help with "Maple" myself. The only thing that is undisputed is that a solution exists and that you describe its structure. As I do not know the aim of your paper, I unfortunately cannot reach a different viewpoint.

@delvin 

Equation (3) is a Painleve equation. It cannot be solved in a closed form. It was converted into a first-order system using (5) and can practically only be solved numerically or by using special functions. There is a lot of new literature on this on the Internet. There is also something about it in "Kamke". Initial/boundary values ​​are required.

@Kitonum 

... to you and dharr 6929. It seems that you both enjoyed solving the problem. I solved it a long time ago using elementary geometric methods. I generally only present problems here that I have already solved myself, so these are not homework assignments that I need help with. Thanks to your solutions, I have now learned how Maple does it. I still need a lot of practice with the Maple commands. In particular, structuring text, command lines and inserting externally generated graphics sometimes cause me problems.

@Kitonum 

... here is a reference to the theorem about the peripheral angle above the chord FC in the circle around D.
(Sorry, I couldn't resist mentioning that.)

@Kitonum 

How does the angle you are looking for change if, instead of the side length of the square, only the property "square" and no side length is specified?
;-)

@vv 

As you might have guessed, proving this statement requires a thorough knowledge of functional analysis and linear algebra. With some effort and literature study, I was able to prove it. Interestingly, this theorem is not included in my favorite books by Riesz/Nagy, Natanson, Hewitt/Stromberg and Shilov/Gurevich. And after further searching, only relatively recent individual articles were found. Their authors are in particular: Pal, Radcliffe and Mirotin. Thanks to your task, I was able to close a gap in my education even at my age. Is there a book on functional analysis/algebra that presents this theorem in its overall context? By this, I mean the structure of the premises and conclusions.

@dharr 

I have occasionally had to solve tasks of this kind when the stability behavior of structures or their natural frequencies/resonances had to be investigated. Unfortunately, the real background of GFY 40's question is not known. And therefore it is unclear whether complex zeros of the calculation model have any significance in reality. Furthermore, the task appears to be a linear model in which "higher order terms" were neglected in the well-known way. I would be interested in information on this. Perhaps the calculated complex zeros are "only" a numerical effect due to the process?

@Rouben Rostamian  

It is impressive that "plot" calculates a table of values ​​(beta; det) in the background and uses it to draw the graphs det=det(beta). I didn't know that and could have used it 50 years ago ;-) . This makes it possible to calculate local zeros of monotonicity arcs with the help of "fsolve" - ​​I am delighted. My suggestion can therefore be seen as a primitive preliminary stage of this procedure and can be forgotten.

@vv 

That is a very interesting task - thank you :-). Since only periodicity is assumed, the concept of function for real-valued functions must be understood more generally. In any case, the statement does not apply generally to "usual" continuous or even differentiable functions. I still need to think about that in more detail, so here is just an attempt at an idea. One solution seems to lie in linear algebra (functions as vector space) and functional analysis (completeness/closure). Let's see if I can come up with something precise. I find tasks like this fun.

@vv 

... cos(0)=1. Therefore (1) a*p=2*m*pi and (2) p=2*n*pi. Inserting p from (2) into (1) and solving for a gives a=m/n and proves the statement according to your idea of ​​proof.
It actually looks as if the statement cannot be checked graphically using the example. It is true that f(x) is periodic for rational a. A continuous approximation of graphs to the state "non-periodic" is probably not possible, although every real number can be represented as the limit of a sequence of rational numbers. Another obstacle is probably the only finite representation of numbers on the computer.
I became aware of this task in 1974.

@Kitonum 

In order to get to know Maple commands better, I am currently interested in the possibilities of checking variants/combinations for target properties. In particular, I would like to be able to examine variants that cannot be easily indexed using a running index 1, 2, ...
How is your code plotted as a result? No drawing was created when it was executed.

@Kitonum 

... for me to practice reading Maple commands in the program. So would you please post the complete code?