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 

restart;

with(plots):

# Define a custom `rmod` function for periodic switching
rmod := (t, n) -> frac(t * n):

# Create a function for the chessboard pattern that switches periodically
f := (t, p) -> if (floor(4 * t/Pi) + floor(4 * p/Pi)) mod 2 = 0 then 1 else NULL end if:

# Plot the black chessboard squares on the sphere
p1 := plot3d(f, 0 .. 2*Pi, 0 .. Pi, coords = spherical, scaling = constrained, 
             color = "black", grid = [200, 200], style = surface, adaptmesh = false):

# Create a white sphere to make the pattern clearly visible
p2 := sphereplot(0.999, theta = 0 .. 2*Pi, phi = 0 .. Pi, 
                 color = "white", style = surface):

# Combine both plots
plots:-display(p1, p2);

If you follow these axioms in Maple , then it is proved
1 is neutral element for multiplication
0 is neutal element for addition 

Tsol := Re(value(eval(rhs(sols), infinity = 12)))

On school we learned also to bring terms to one side ...:-)

ode := diff(u(x), x, x)/u(x) - diff(u(x), x)^2/u(x)^2 = x^2 - diff(u(x), x)^2/u(x)^2;

# Move all terms to one side
eq := lhs(ode) - rhs(ode);

# Simplify the result
simplified_eq := simplify(eq);