@Rouben Rostamian  

Whether a plate can behave like a membrane strongly depends on the choice of material, thickness, and the forces it is subjected to. A thin, flexible plate made of materials such as rubber or plastic can indeed exhibit behavior similar to that of a membrane when under tension. However, for stiffer materials, such as metals, the bending stiffness is often dominant, meaning they generally cannot function as a membrane so easily.

If you take a steel plate or a plate what acts as a membrane it s not clear , so i decided to go for the more easy case 

@Rouben Rostamian  , Thanks , for correcting this

"Maple's pdsolve,numeric is designed to solve PDEs in two independent variables.  So we are out of luck with solving for w(x,y,t)"

Indeed i remember it now again .itis only numeric solving  for x ,t 

Note: i am using AI with Maple, but "Maple coding expert" chatgpt seems to be ou of the air  unfortunaly
I am forced to use the matehematica's  AI, and this give code for numerical solving pdes ..it is not clever enough for analyzing rmaple code as it seems :-)

@dharr  By the looks of it, the Sturm-Liouville theory does not work for nonlinear odes unfortunately I read.
It would be too good to be true that all odes types can be solved with it. 
The theory does work for linear odes and some type of pde as it seems 

Maybe in direktmath ; drag and drop 
Also possible this functionality in Maple somewhere, but a very basic level ? 

Download DirectMath

It is primarily not about the solution of the ode to write it in a standard form (interesting to know, though, how it is done here).
It is about the ode equation to write it in a kind of standardized form, as it is also used in mathematics literature to make it easier to read.

” A canonical form is a standardized or simplified representation of a mathematical object or structure, such as a matrix, equation, function, or number, giving it a “standard form.” By expressing an object in its canonical form, it becomes easier to solve equations, analyze properties, and compare objects, as similar types of objects have the same structure in this form.”

@dharr 
 

Thanks, this classification looks professional and don't know if this is how it is used worldwide?
Whether this is a true mathematical standard form used in mathematical literature is not known to me. ( am not an expert )

The mathematical notation is not always unambiguous in many cases.
I found another overview , but there are elaborations attached to it and that is what I am concerned about.
Here is a part attached as an example notation 

@Alfred_F 
Thanks, yes you have corrected now  your mistake what i noticed earlier, good.
"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."
I see there exists a compendium for non-limear odes, maybe i can track the ode there too ?

If you own this book it seems to me, then you can look for solutions to odes in a more structured way and not have to reinvent the wheel every time

Found another resource for ode solving : 12000.org/index.htm

I'm not a big fan of deleting a question.

Often repetition can make a topic clearer for the questioner and also for the person trying to help .
The ideal case is for the questioner to continue with the question they are working on and ask new questions about it , until some things become clearer to the questioner .

This does require patience on the part of the person helping and a helpful attitude

This is how @salim-barzani' s question was put away and I wanted to respond to it further and had responded to it, well..

@salim-barzani 
Copy the code and paste then in a empty worksheet and it will work when you run it
Easy going.... 

Is it for a 1Dimensional  wave equation or 2 Dimensional wave equation this soliton wave?
Note: i am a beginner with this 

@salim-barzani 
Probably is this the most compact expression you can get, but useless because you don't know the values for A and B, belonging to the solutions 
 

restart; with(PDEtools): with(DEtools):
sol := [seq(dsolve(eval(diff(G(xi), xi) = G(xi)^2 + A*G(xi) + B, {seq(x = 0, x in s)})), s in combinat:-powerset({A, B}))];

with a set calculation  ( set A and set B )
 

sol := [seq([s, dsolve(eval(diff(G(xi), xi) = G(xi)^2 + A*G(xi) + B, {seq(x = 0, x in s)}))], s in combinat:-powerset({A, B}))];

The power set of a set A consists of all subsets of A.
The power set of is written as P(A) 
Proposition: If set A has n elements, then P(A) has 2^n elements
P(A,B) ....

@salim-barzani Can you use deepl for translating your language into Englisch ?
Vertalen met DeepL Translate - 's werelds meest accurate vertaler

@salim-barzani 
Explain what exactly you want to compute about an pde and whether that is a repetitive computation ? 

With odetest