@salim-barzani 
Its possible to get your expresssions in the wanted form (the art of expression massage :-))
https://www.mapleprimes.com/questions/238929-How-Simplify-This-Expresion-In-Maple
This former thread is a nice example how you can study this art and become better

I did a small part for eq_8  only and made also a procedure for it 
Look how i split the commands and i am using also a AI maple coding expert for questions if they com e up.. 
You have to be creative and make an analysis of an expression and discover something mathematical in it  
I did just a little begin.
This_way_mccdarr_ode_manipulatie4-9-2024Mprimes.mw

myODEProcedure := proc()
    local eq_7, negsol, CS, replace, B1B2, eq_8;

    # Define the differential equation
    eq_7 := diff(G(eta), eta$2) + sigma*G(eta) = nu;

    # Solve the ODE under the assumption that sigma < 0
    negsol := rhs(dsolve(eq_7) assuming sigma < 0);

    # Display the intermediate result for the original solution
    print("The intermediate result for negsol is:", negsol);

    # Convert the solution to a trigonometric form and expand
    negsol := convert(negsol, trig);
    negsol := expand(negsol);

    # Display the intermediate result after trigonometric conversion and expansion
    print("The intermediate result for negsol after trigonometric conversion and expansion is:", negsol);

    # Define the replacements for hyperbolic functions
    CS := [C, S];
    replace := convert(indets(negsol, function), list) =~ CS;

    # Display the replacements that will take place
    print("The replacements for hyperbolic functions are:", replace);

    # Replace cosh and sinh with C and S and collect terms
    negsol := collect(eval(negsol, replace), CS);

    # Display the result after replacing and collecting terms
    print("The intermediate result for negsol after replacement and collection of terms is:", negsol);

    # Solve the coefficients in terms of B1 and B2
    B1B2 := solve({coeff(negsol, C) = B1, coeff(negsol, S) = B2}, [_C1, _C2]);

    # Display the result of the coefficient solution
    print("The solution for the coefficients B1 and B2 is:", B1B2);

    # Create the final equation eq_8 by substituting the found solutions
    eq_8 := eval(eval(negsol, B1B2[]), (rhs = lhs)~(replace));

    # Return the final equation
    return eq_8;
end proc:


eq_8 := myODEProcedure();

@salim-barzani 
What book you are using ?,then i can do more ( i hope:-) )
 

 
EXPLORATIONS OF MATHEMATICAL MODELS IN BIOLOGY WITH MAPLE™
MAZEN SHAHIN Department of Mathematical Sciences Delaware State University

Download Module_VanderPol_sytemsexpl_2-9-2024_Mprimes_animatie.mw

size on 800,800 

@Paras31 ,Thanks
Try out some values for mu : 1/2,1,3/2 ,3 

Module_VanderPol_sytemsexpl_2-9-2024_Mprimes_module.mw

Simplify it for radians and look what is left :-) 
2Pi radians = 360 degrees 

 

I  like to see a better fieldplot and used dfieldplot from DEtools
No vector procdedure  input anymore, but a system of ODe's



 

Download DDA_procedure_omgezet_31-8-2024_Mprimes_ODEsyteem.mw

If you want to see flowlines of a system ode's 

 

Download DDA_procedure_omgezet_30-8-2024_Mprimes.mw
 

	
#new_sol
new_sol:=lhs(sol)=evalindets(rhs(sol),'specfunc(anything,Sum)',X->Sum(expand(op(1,X)),op(2,X)));

I don't understand this, but it works 
There is also a another approach,but for that you must know some sum rules  

@nm 
Thanks,
This is the right general solution. 
Note: the first solution pic was wrong, because i copied not the whole expression (scrolling not complete )
Note 1 did not use a book text, but ask ai 

@nm ,  thanks , good idea to add small code directly

and c is wave speed 

Impressive ...
A good exercise to do (try ) this in geometric expression app

Lotka_Volterra system as start for experimentation with bifurcation 
Note: the spiral must be bigger ..
IterativeMaps:-Bifurcation ..use?
I do have extensive old  Maple studymateria from my education  for Lotka -Volterra , but it is not using ODE's, but works with difference equations, to study the phaseplot with equibrillium lines


lotka-volterra_bifurcatie_analyse_test_-25-8-2024.mw

Dynmod08_-_kopie.mws

@Paras31 
Getting a good plot in Maple is still a challence as a non-expert as i am.
You have done well.

@Paras31 
 

systems := [
    (x, y) -> y, 
    (x, y) -> -delta * y - alpha * x - beta * x^3 + gamma * cos(omega * t)
];

Conclusion:

For the given system with a time-dependent forcing term gamma * cos(omega * t), there are no true equilibrium points in the classical sense because the term gamma * cos(omega * t) is time-dependent and prevents the system from reaching a stationary point. 

Putting the system in the right input format of the Dsys procedure ,then you will probably see this phaseplot?
For t = 10 , this is a solution curve