How can I solve this non-linear ODE exactly or approximately in some series solution?

Non_linear_ode.mw

the Initial value problem is well defined but i could not find solution of the problem. A number of method have been used but all are useless. How to obtained graph of the all equations. Also can we find the value of R (radius) where P is zero.coupled_IVP.mw

right_file.mw

i want to start i from 0 ,1 ,2 , 3 but it does not work I rhink the defaul value is from 1 to 4. can we begin it with zero.

A lot of efforts I performed to obtained the right results, somehow I remain sucessful but the results are not exactly same as expected. I request you to kindly see at the christoffel symbol. I also sent the right results.

some_mistakes_in_christoffel.mw

you can see the right results in the given below attachment

right_christoffel_symbols.pdf

as you can comprare the results of both file. the obtain crystoffel are not same as the requied.

How can we find ricci tensor of a metric attaced in given file that involve two unknown functions phi(r) and nu(r).metric.mw