@salim-barzani 
I understand, it is not consistent behaviour the moderator is showing, so is it yes or no ?
Had he moderator kept reminding you to do things differently with the question , you would have done the right thing I think.
The moderator also let it happen and didn't get on top of it to correct you and now just delete a question.
This does not deserve any beauty prize and the moderator could have put your wrong post under the existing question as an example and reminded you again of the correct procedure to ask questions.

This moderator falls short in didactic quality here

So add your deleted question /post  complete again here to this existing question row or as a branch of the existing question,then you fullfill to the rules

@salim-barzani 
yes, when it is the same topic closely related , its better to keep it here on the forum as a whole 
So , add again the complete ! deleted post  to the row of existing answers from How define D-operator (bilinear hirota derivative) for PDEs?
Or make a branche then from this new question 

@mmcdara 
"'m sorry but the font is too small and the text too fuzzy for me to see clearly what it's written" 
In this case you can zoom in /out with middle mouse scroll wheel,  to read better on the page on windows
But this is not the complete @removed post ..?

@mmcdara 

Where is the other most recent question about this topic ?
It seems to be deleted, by whom..
As user are removing their posts , when it is answered by a Maple specialist here on the forum, then it must not be deleted

@salim-barzani 
Is there now availability of two external Maple packages, which can support this search ?

@salim-barzani ,Whether there is a true pde specialist to be found here on the forum, who masters all the techniques to manipulate the pde's ?
If not, then there are forum members who are a specialist in the maple language and master it just fine, but are not aware of the techniques used with PDEs 
The only thing left then is to describe those desired techniques on the forum in the hope that a maple specialist can translate it into maple code.

But maple specialist help is much easier via chatGPT to get an idea of the techniques used with PDEs
I also read some of this way about the bilinear form of a pde, but this requires study of the topic and context. 

@salim-barzani , What makes@dharr not arrive now at the same result as in the document ?
I remember there was an issue before a while back , in which a result found still needed some ‘formula massage’, before it could be seen in the required format as the document read.

Give more details maybe someone can answer this?

Root two is an irrational number and you take this value for a
If you then assume that x is a real number. 
Then what will maple calculate for the function values ?

@scallopedpancake 
Setup screen ? in Physics package 

As a system of odes via  ODEsteps : lambda and mu  , for symbolic solving  there are no solutionsteps , but for lambda and mu = 1 there is a solution
Note : output odesteps not correct ?


 

Download Find_ODE_solution_with_condition_parameter_als_syteem_via_odesteps_26-11-2024.mw

@dharr  "avoids square roots of negative numbers in practice" . This is calculated in the complex number system.
 

restart;

# Define the coupled differential equations
de1 := diff(F(eta), eta) = lambda * G(eta):
de2 := diff(G(eta), eta) = mu * F(eta):

# Solve the differential equations (general solution)
sol := dsolve({de1, de2}, {F(eta), G(eta)}):
sol_hyperbolic := convert(sol, trigh):  # Hyperbolic form

# General solution (hyperbolic, valid for all lambda and mu)
print("=== General Solution (valid for all lambda and mu) ==="):
print(sol_hyperbolic);

# Alternative solution for lambda * mu < 0 (explicit complex root)
assume(lambda < 0, mu > 0):  # Assume lambda * mu < 0
complexRoot := simplify(sqrt(lambda * mu)):  # Explicitly compute complex root
complexRoot_imag := I * sqrt(-lambda * mu):  # Write root in imaginary form

# Replace hyperbolic functions with trigonometric functions
sol_trig := subs(sqrt(lambda * mu) = complexRoot_imag, sol_hyperbolic):
sol_trig_converted := simplify(sol_trig):
print("=== Alternative Solution (Trigonometric Form for lambda * mu < 0) ==="):
print(sol_trig_converted);

# Add information from the image: interpretation and cases
print("=== Interpretation of Cases ==="):
print("Case (I): Covers all combinations of lambda and mu signs using hyperbolic functions.");
print("Case (II): If lambda * mu < 0, the solution can be rewritten in trigonometric form.");
print("Note: Hyperbolic functions naturally handle both positive and negative values of lambda * mu.");

# Validate both solutions with odetest
validation_hyperbolic := odetest(sol_hyperbolic, [de1, de2]):
validation_trig := odetest(sol_trig_converted, [de1, de2]):

print("=== Validation of Solutions ==="):
print("Validation of the hyperbolic solution:"):
print(validation_hyperbolic):  # Result must be 0
print("Validation of the trigonometric solution (lambda * mu < 0):"):
print(validation_trig);  # Result must be 0

No complex numbers anymore or do you mean something else ?

@Carl Love 

Yes, some explanation of the code can never hurt in my opinion.

But to really understand maple code better, the AI is also a good helper, but paid via ChatGPTplus
Unfortunately the Maple coding expert does not seem to be around anymore, so it will be Wolfram or yet other chats .

@Aung 

Then you should rewrite a general expression of the current integral expression into a general form of an integral by parts ?
 =   ( with partiele integration )

Both expressions i let check this  with  parameters and they seem to be numeric equal