@GFY 

My initial guess that theta_3 and theta_4 are free to chose was incroeect. I striked it through in teh above. What describes it better: Theta__3 does not depend on theta__4. I.e., for a valid theta_3 there are two valid values for theta__4.

When you plug in the above 16 solutions for a_1 and c_1 into the 4 equations and solve for theta_3 and theta_4, you will get all combinations of variables (in total 64) that are solutions for the system of equations

Here is the first solution for theta__3 from equation 1

subs(a__1 = 5.454567591 + 0.005714161093*I, c__1 = 0.002689473223 - 3.665495195*I, secular7[1]);
solve(%, [theta__3]);
      /               12                 9  \            
      \-1.888144781 10   - 1.978005270 10  I/ ((0.0063172

         - 0.1217592075 I) cos(theta__3)

         + (0.121759209 + 0.006316869337 I) sin(theta__3))


          [[theta__3 = -0.8090474571 + 6.755468571 I]]

which combines with two solutions for theta_4 from equations 2 and 4 to

[a__1 = 5.454567591 + 0.005714161093*I, c__1 = 0.002689473223 - 3.665495195*I],[theta__3 = -0.8090474571 + 6.755468571*I],[theta__4 = -0.7836748989 + 8.209192730*I];
[a__1 = 5.454567591 + 0.005714161093*I, c__1 = 0.002689473223 - 3.665495195*I],[theta__3 = -0.8090474571 + 6.755468571*I],[[theta__4 = 0.7871214279 + 8.209192730*I]]

For equation 3 you get in the same way 2 more equations sets of solutions

@NIMA112 

Here is the idea for H[1] which computes much faster. Carefully check if the expressions were copied correctly. C must be part of the sum since it contains n. Check if that is really what you want. The resutling expression is still big.

ABC.mw

No error with 2023.2

@GFY 

Below is a way to solve for the 16 roots. As said above theta_3 and theta_4 are free to choose. If you replace the floats by parameters or rational numbers, this solution will be exact.
 

Download solve821_reply.mw

Have you checked for backups?
Depending on how Maple is terminated there should be backups if the options are configured like this

I thought when odetest returns zero without assumptions then the tested solution is valid over the complex domain. In your example this is not the case.

The worring consequence is that plotting for a given parameter the solution of one of the "root ode"s (where the residual of the identical solution of the origianl ode is non zero) does not show an empty plot. For this reason, I am now also of the opinion that one should work with the odetest of the original ode.  There is a flaw in my reasoning that no assumptions are made when using PDEtools:-Solve that I would like to understand.

I always see the input window with Maple 2024.1 and standard GUI settings. The window is not moving. Same with Maple 2020. What do you use?

Did it look like this?

It took a while, then it worked.

Hnx said he cannot reply

@nm 

You also use exponentiation as I did in my verification. I would be interested if a knowledable user could comment on the exponexponentiation operation. Does it "cut of branches" or is it a general operation that can be done without any restrictions?

This is too complex for me.

@ecterrab 

Your answer clarifies allot. It describes in general how dsolve works and why there can be many solutions. I was looking for something like that.

My answer was only guess work, based on observations I made here and there. I did not delete it after sending when I saw your answer (to document at least where a user is right or wrong in his interpretation).

I would not have guessed to find something of interest in ?dsolve,setup. So I never consulted this page. Besides technical details I now can describe myself better as a curios standard user. However, this page rasies again the question about the difference between classificon methods, returned results and methods for ODE solving. More specifically, why is dAlembert not listed under "Methods for 1st order ODEs"? Should this be clear to the standard user?

Finally, some thoughts on PDETools. PDETools is mentionned (not only on the dsolve,setup page) in a way that it is a natural part of ODE solving. Many answers here in this forum use PDETools as well. Why should a standard user interested in ODE solving consult PDE librararies? I have not found an answer to it yet. For me, the use of PDETools sounds like a workarround for something that is not possible with ODETools. If a libraray function can be used for PDEs and for ODEs shouldn't it be part of a DETool package?

Anyway, exanding at bit more on how dsolve works and the existence of branched solutions (I doubt that many users are aware of this) could strenghten the educational aspect of Maple in the nontrivial domain of differential equations. Here Maple is my first point of reference since I want to use Maple to get solutions. The other way arround (studying ODEs with a textbook and then finding the right commands in Maple) does not make sense to me.

Maybe also of interest:

https://www.mapleprimes.com/posts/220633-DMS-Degrees-Minute-Seconds-To-Decimal-Conversion

Not clear what you want to simplify. You could combine the roots under certain assumptions.

combine(sqrt(2)*sqrt(a[5])/sqrt(a[4])) assuming (0 < a[5], 0 < a[4])

@Thomas Richard 

That works. The tricky bit is when the spacecurve is not an explicit function of time and generated by numerical integration. If I understand @mmcdara solution correctly, he solves a sequence of IVP where each solution generates the initial values for the subsequent IVP. Quite some coding required for numerical solutions.
Thank you

@mmcdara 

That is a step in the right direction. The colorsheme option is of value when the animation is stopped.
Thank you!