@mmcdara 

So the small orthogonal lines mean that all sides are the same length. That was new to me.

According to Picard Lindelöf's theorem, the interval (approximately lying at) -0.5<x<1 must be considered separately. Here there are discontinuities on the right-hand side of the ODE. Continuity and the existence of the solution are only guaranteed outside of it.

What do the short orthogonal crossbars on the polygon sides mean? I don't know this symbolism. Are the polygon sides all the same length? In this case, the task would be easy to solve. Otherwise, at least ABC forms a kinematic chain. Further specifications would be required for a clear solution.

My first answer needs to be corrected. I didn't notice that the ODE was presented in an abbreviated form in the posting. I therefore missed the square of the first derivative.

...there under number 2.422 on page 504 you will find the solution: y=A*cot(x)+B*(1-x*cot(x)). Substitution according to Riccati y´´=y´*u could help.

@vv 

... and I wish you a peaceful Christmas.

@vv 

I know this element-wise proof. I had hoped to find hints for a different solution in the "Group Theory" package of the help text. To do this, it would be necessary to use the symbolic calculation rules for regular matrices to solve the equation A*B = B*A step by step for B without element-wise calculation. In this, B is the matrix we are looking for. One approach to this would be B-A^(-1)*B*A=0 and

det(B - A^(-1)*B*A)=0. With x for the eigenvalues ​​and regularity of A^(-1)*B*A, det(E*x - A^(-1)*B*A) = 0 would also be possible. From this we can guess the well-known result.

@Kitonum 

The solution is the period of the fraction 1/7.

@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.)