Numerical solution of an ODE using NDSolve

Question

I am to find the solution $R(\rho)$ to the following differential equation,

$\frac{\partial}{\partial\rho}\left(A^2\frac{\partial}{\partial\rho}R(\rho)\right)-l(l+1)R(\rho)=0$

where $A$ is a numerically inverted function of $\rho$. My code is,

 rho[r_, b_, q_] := (2 b/(1 - q)) (1 - (b/r)^(1 - q))^(1/2) Hypergeometric2F1[1/2, 1 - 1/(q - 1), 3/2, 1 - (b/r)^(1 - q)]
 Tp = InverseFunction[Function[{r, b, q}, rho[r, b, q]], 1, 3];
 ode = {D[(Tp[\[Rho], 0, -1])^2 D[R[\[Rho]], \[Rho]], \[Rho]] - l (l + 1)R[ [Rho]] + 2 \[Xi] a^2/(\[Rho]^2 + a^2) R[\[Rho]] == 0};
 Bc = {R[0] == 1, R'[0] == 1};
 rules = {AccuracyGoal -> Infinity, PrecisionGoal -> 20, WorkingPrecision -> 30, MaxSteps -> 10000};
 Rsol = R /. First@NDSolve[Union[ode, Bc], {R}, {\[Rho], 0, 10}, rules]

This is what Mathematica says: I don't know what is wrong with my code.

Answer 1

It is necessary to express the derivative of the inverse function in terms of the derivative of the original function using the rule $dy/dx=1/dx/dy$:

f = InverseFunction[
   Function[{y, b, 
     q}, (2 b/(1 - q)) (1 - (b/y)^(1 - q))^(1/2) Hypergeometric2F1[
      1/2, 1 - 1/(q - 1), 3/2, 1 - (b/y)^(1 - q)]], 1, 3];

f1[x_, b_, q_] := (
 Sqrt[1 - (b/f[x, b, q])^(1 - q)] ((b/f[x, b, q])^(1 - q))^(1/(1 - q))
   f[x, b, q])/b
ode = 2 f[x, b, q]*f1[x, b, q]*g'[x] + f[x, b, q]^2*g''[x] - 
    l (l + 1) g[x] == 0;
b = 1; q = -1; sol = 
 ParametricNDSolveValue[{ode, g[1] == 1, g'[1] == 1}, 
  g, {x, 1, 10}, {l}] 
Plot[Table[sol[l][x], {l, 0, 5, 1}], {x, 1, 10}]

Note that in a certain range of parameters the solution becomes complex, so you need to print the real part or the absolute value of the function