Doubt about how to mathematically formulate my problem

Questions
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1

I’ve been working on my master thesis and faced with a big problem. I don’t know how to formulate this problem in mathematica.
I’m working with modified TOV on Jordan Frame and i have three equations to solve

\phi[r]= 2\alpha R[r]+1

\lambda'= \frac{1}{\left(2r\phi+\phi'r^{2}\right)}\left\{ \exp\left(2\lambda\right)\left[k\rho r^{2}-\frac{1}{8\alpha}\left(\phi-1\right)^{2}r^{2}-\phi\right]+\phi+2r\phi'\left(r\nu'+1\right)+\phi''r^{2}\right\}

\nu'= \frac{1}{\left(2r\phi+\phi'r^{2}\right)}\left\{ \exp\left(2\lambda\right)\left[kPr^{2}+\frac{1}{8\alpha}\left(\phi-1\right)^{2}r^{2}+\phi\right]+2\lambda'\phi'-2\phi'r-\phi\right\}

where \lambda, \nu, \phi are function of r.

I have three conditions
$\nu[0] =0 \
\lambda[0] =0\
\phi[0]>0$

how can I fix my problem?

Suppose that the EoS is
P[r] = k * rho[r] ^2

Open in Wolfram Cloud ↗
2

In your notebook that you attached, thank you for that, would be impossible to help otherwise,

1: You have M[r] and P[r] and R[r] but you are not solving for those and have not defined those.

2: In eq3 you use := and = which should perhaps be = and ==.

3: eq1 doesn’t appear to include a derivative, but I don’t know what R[r] is.

4: eq2 has rphi[r] which should probably be r*phi[r] and rnu’[r] should probably be r*nu’[r]

5: eq4 has rphi[r] which should probably be r*phi[r]

6: In NDSolve you have phi’[0]=1 which should probably be phi’[0]==1

See if you can fix all those and attach your new improved notebook to the bottom.

With that we can start looking for the next level of problems to work on