@Scot Gould 

Thanks, I'll try it. Please answer this question:
How do I delete an empty/unused input line for execution commands that is located anywhere in the worksheet?

@vv 

"The polygon with maximal area is the cyclic one" - that is not obvious. More details in Lit.:
[1] Mathematics Magazine, Vol. 39, 1966, issue 4
[2] Geometric Problems on Minima and Maxima, Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov, Birkhäuser, Boston Basel Berlin

Additional question:
The circumcircle has a center. Which of the polygon vertices, ordered according to increasing length of the sides, are the vertices of the triangle whose perpendicular bisectors intersect at the circumcircle center?

@nm 

You're right. I should read more carefully :-(.

@Carl Love 

...not the calculation of the second initial condition for y´(0). Rather, the recognition that differentiation of the original equation to "deq" and subtraction of "eq" makes the integral disappear is important as training for future solution ideas - what remains is an ordinary differential equation that is easy to master with the help of Maple. Using Maple is, after this realization, simple, understanding "handwork" :-).

@mmcdara 

It is enough to calculate deq minus eq and the integral is gone. The task doesn't require any more tricks.

@mehdi jafari 

... us have a look ;-)

@Kitonum 

Yes, there is power in these Maple commands. In the past, the formation law had to be laboriously determined using the characteristic equation (literature e.g. Markuschewitsch, Lin. Rekursionen). It is important that pupils and students learn this background-knowledge in order to be able to use it if the computer breaks down ;-).

@salim-barzani 

...wavelets? I have no experience with solitons in solving methods. But as far as I can remember, there are methods that use wavelets, e.g.:

https://ijsts.shirazu.ac.ir/article_2153_836c1c7bda66ef58fcead78d7a6f9f97.pdf

Maple offers something about this in the help.
Good luck!

@salim-barzani 

..., but I can't help you with that. I was only interested in solving your equation out of curiosity in memory of "old times" even without knowing your specific physics background.
Finally, a link:
https://www.colorado.edu/amath/sites/default/files/attached-files/2015_uc-london.pdf

@salim-barzani 

I wrote Painleve Type II! A square of the first derivative is not included in this equation structure.

You want to calculate the zeros of a 4th degree polynomial. This is possible in principle, but results in terribly long formulas (https://en.wikipedia.org/wiki/Quartic_equation). It would be much easier to treat each specific case numerically.

The vibration behavior of thin elastic plates is probably to be investigated. To do this, the geometry/contour and the bearing/support as well as material properties must be specified. The plate behavior is then described by a well-known so-called plate equation (partial differential equation). Do natural frequencies have to be determined or is there an excitation load? Is damping present? There is extensive literature on this. I recommend "Werner, Structural Dynamics" page 146 and "Szabo, Advanced Technical Mechanics".

You should write this down on a worksheet to start with. Then we will move on and solve the equation.

The conditions of the Picard-Lindelöf theorem are crucial for the existence and uniqueness of an explicit solution to the given differential equation. In the present case, under the current initial conditions, these are only fulfilled in a rectangle of the x-y coordinate system with approximately 0<=x<62 and -18<y<57. Only in this field does a consistent direction field exist. As a Maple newbie, I don't know how to plot an implicit function, so I did this with the help of good old derive. If you leave out this rectangle restriction, the graph is obviously ambiguous. Maple probably has problems with this and therefore does not allow a solution for y(x).

@janhardo 

Yes, but that doesn't make the solution any easier. Only the numerical approach will probably lead to the goal.

Here are my solutions: