I can obtain a closed form solution for the first equation using the Wolfram Mathematica, however there is a problem with the second equation. I will let you know about my results by Monday.

Provided that a closed solution exists, you can also try Maple to solve symbolically.

I can obtain a closed form solution for the first equation using the Wolfram Mathematica, however there is a problem with the second equation. I will let you know about my results by Monday.

Hello Dear Mohammad Amin Motamedi ;

Thanks for sharing me your ideas. I am following up some videos on perturbation method. So your suggestion is for me to use Method of Multiple Scales? There are some different approaches on Perturbation method, but which one suits my case? Any ideas on how to start?

Hello Dear Filippo Maria Denaro

I tried to use mapple vers.2019 to solve my equations but its returning me implicit answers in Z domain and in integral. I dont understand what these are. Please see the attached file.

Dear Janusz Pudykiewicz

Awaiting your valuable reply.

Dear Caner Beykont

The analytical solution obtained by Wolfram Mathematica can be found in the file: analytical_solution

I also attached an example of a numerical solution and the corresponding plots (also from Mathematica) in the file: numerical_solution

I hope this is relevant.

Dear Janusz Pudykiewicz

I checked the results, thanks for the time. I will need to C-Code this result into a microprocessor. So I need to understand several things in the answer. Can you support me on this?

1. What is Cos[#1]

2. How do I C-Code EllipticE function?

3. There is an & sign at the end, what does that stand for?

4. This is overall an inverse function. So I suppose I should use Newton's method or any-other search algorithm to find the result of the real function. How do I program this inverse logic?

So in summary I need to convert this result into a C-Code language. Maybe there is an easy conversion method from Wolfram to C?

Thanks for your kind support up to now.

Dear Caner Beykont

You are most welcome. Some additional suggestions are as follows.

Conversion of the Mathematica script to a program written in C is discussed in the following links

https://reference.wolfram.com/language/tutorial/GeneratingCAndFortranExpressions.html

and

https://reference.wolfram.com/language/CCodeGenerator/tutorial/Overview.html

Concerning the concept of an inverse function in Mathematica please consult

https://reference.wolfram.com/language/ref/InverseFunction.html

Before converting the solution to C it is advisable to investigate the functions playing with the formulae using the traditional method (pen and paper) The numerical solutions are also useful as the additional method of verification.