@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:

@mmcdara 

The solution deserves several thumbs up. Where can I type that in?

@Rouben Rostamian  

That is great :-)

@vv 

One solution would be to calculate it in detail ;-) . Question: How long is a coastline that is being measured in ever greater detail? What happens with polygon in lim ---> oo ?

@Carl Love 

No rings - or topologically equivalent figures of higher genus, otherwise everything is allowed.

@janhardo 

Maple and Mathematica cannot find a solution in your source either. Without special transcendental functions, it will generally not work. For this:
https://www.12000.org/my_notes/solving_ODE/current_version/chapters/Differential_Gleichungen_E_Kamke_3rd_ed_Chelsea_Pub_NY_1948/indexsubsection1585.htm
7.11, 1601, #10565

@janhardo 

The "wheel" for this topic was already described in 1902 and 1926 (Painleve, Ince). A reference to a general overview of the treatment of ordinary differential equations is of little help. The fact is that the ODE in question can only be solved in closed form in special cases. More precise information on the coefficients is required.

If I understand your question correctly, you are investigating a nonlinear ordinary differential equation of the structure
a*y´´+b*y^3+c*y=0.
Equations of this type are discussed in E. Kamke's book "Differential equations, solution methods and solutions", volume 1, from page 544. The references to further literature are interesting. According to this, so-called closed solutions only exist in special cases. For practical cases, the "numerical club" is usually required. And this is easily possible if a system of two first-order equations is generated from the original equation.

@one man 

"I really like moving pictures." I like them too. Unfortunately, as a Maple newbie, I am not yet able to create a worksheet with animated graphics myself. Do you know Holditch's theorem? An animation of it could help learners, for example, to understand the thinking behind Guldin and Cavalieri's rules as an introduction to the concepts of "integral" and "topological product".
Would you give me Maple help getting started with an animation?

Thank you for this article. It reminds me of the beauty of projective geometry, which is now easier to calculate with the help of computers. Such interesting theorems, which make the duality "... lie on ..." and "... go through ..." visible, are illustrated in a way that is easy for learners to understand. Last but not least, I am also thinking of Desargue's theorem ;-) and a "difficult" task posed here.