@Rouben Rostamian  Thank you for your help.
One issue is how can I introduce variable  into the operator ? In your ode, there appears D, D^(2), D^(3), etc. which are not clear for me what they represent when taking the derivatives of .

@dharr Thank you.
Yes, that hint helps but I was wondering if I could find a solution in the form of  times .

@Carl Love Yes, that is the problem that I have and wondering how can I get rid of this issue.

@ahmadtalaei 

Thanks a lot for your help.

@dharr 

Thank you so much. Mine is 2016 and that is probably the issue. Anyway, could you please convert two hypergeom functions to given in the attached file to the LagendreP?

Download convert-Legendre-v1.mw

Download convert-Legendre-v1.mw

@dharr 

I was able to upload the file. Thank you.

I find on the following page where these two functions can be converted to each other. Please see equation 14.3.1 in the following page:

https://dlmf.nist.gov/14.3

But the problem is that I couldn't find value for ν in the hypergeom function F(ν+1,−ν;1−μ;1/2−1/2x) that could result in the Legendre(ν,μ,x) where x=cos($\psi$).

@Mariusz Iwaniuk 

I always get a problem copy&paste from maple. Here is the link provided to the question:

https://math.stackexchange.com/questions/3254765/how-to-convert-a-hypergeom-function-to-the-legendre-function

@Thomas Richard 

Thank you for your comment.

Could you please follow the answer by @tomleslie in below which attaches a Maple file and help me to find a way to solve this PDE?

@tomleslie Thank you.

I thought the separation of variables could help. But do you have any suggestion on how to solve the PDE?