@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();