@nm 

Hi,

I thank you again.

I found that there is a difference between your two solutions; in your first solution, the photo sent to me the other night, there was not the function of InverseJacobiAM, but your second solution did include it.

Please send me your code in Mathematica so that I could run it in my Mathematica software. I really need the file.

Best,

@Kitonum 

Hi,

I thank you for taking your precious time. I do not have any specific values for the parameters!

@nm 

Hi,

I thank you for the simple and great answer. I just entered the same code as you did( In[1]:<<Rubi`) but I faced an error!

Please send me the file to run it in my laptop.

@Preben Alsholm 

There is a physical meaning for the relation between C1 nad B in Mechanics. I do know that 'dsolve' works for differential equations, however, the desire result could be derived by it!

@tomleslie 

I meant why it was ok and done in your computer but not in my laptop. How did you edit the file? For example: alpha1 was changed to alpha[1]...
Did you write the whole file again?

@tomleslie 

Hello, yes it's ok now. It's ambiguous yet.. What have happened?

@Kitonum 

Thanks. But it doesn't work for me.... It drove me crazy!!! It workes correctly last week. I had to get some results by the file today...

@tomleslie 

Great, God bless you and happy new year.

@tomleslie 

What a simple solution!

Thanks.

@_Maxim_

The integral term in F2(r) must be a value necessarily as long as those two Nu are close toghether. Indeed in the loop I want this expression " Error= (Nu_final- Nu_guess)<1e-3" to be satisfied and then the loop is going to be interrupted.

The program is correct in the first iteration for \phi= 0.06 and "initial guess Nu=3.6941" and the "final calculated Nu= 3.6942". I already know the Final Nu for \phi= 0.06!!!! But for other values of \phi I could not get any solution precisely.

@_Maxim_

By the way, it does not matter whether or not k is zero @ initial guess "For The First Time", the first iteration in a potential loop. As you know the initial guess for Nu is just for calculation of the integral and after that I removed it by this command: Nu:= 'Nu'.

The only important thing is this fact that the final calculated for Nu, in the end of the loop, must be equal to the initial guess.

@_Maxim_ 

Hello, thanks for your consideration dear.

Yes, I want to find the Nu if the determinat is equal zero. And the problem is exactly here that the at the initial guess the determinant, k, might be any number, not necessarily zero. That is why I want to correct the Nu by replacing the initial guess with the final calculated Nu. It assumed to be difficult for me.

@vv 

Thanks. I must change the domain then! That's 3.195986267 if

0.2 <r< 1

@Markiyan Hirnyk 

Thanks again.

@Markiyan Hirnyk 

Thanks dear,

Yes you're right. I solve that system in Matlab (written below) and those solutions are the same. 

[ 1.6759649064951345004378236436225, 0.7729562969870007822477027324705, 1.9940322524732747423780039106827]

By the way, I wrote some indices wrongly in my question!!! Now I corrected it and we can see this is true as well.  

Download Edited_Solve_Equation.mw

Best.