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I have a differential equation which looks like:

r'[l] = Sqrt[g11[r[l]]*gtt[r[l]] - g11[r0]*gtt[r0]]/
Sqrt[grr[r[l]]*gtt[r[l]]*g11[r[l]]]

I want to find the r[l]. I tried using DSolve

DSolve[{r'[l] == Sqrt[g11[r[l]]*gtt[r[l]] - g11[r0]*gtt[r0]]/
     Sqrt[grr[r[l]]*gtt[r[l]]*g11[r[l]]], r[0] == r0}, r[l], l]

But it is taking very long for my computer to solve. If anyone has any idea how to solve such differential equation, please let me know. Thanks.

My values for all the term:

$Assumptions[Inequality[r[l], GreaterEqual, r0, Greater, 1], Element[{a, B}, Reals]]
gtt[r_] = (-r^2)*Exp[2*As[r]]*g[r]; 
g11[r_] = r^2*Exp[2*As[r]]*h[r]; 
grr[r_] = Exp[2*As[r]]/(r^2*g[r]); 
g[r_] = 1 - (Exp[(3*a - B^2)*(1/r^2)]*(3*(a/r^2) - B^2/r^2 - 1) + 1)/
     (Exp[(3*a - B^2)*(1/rh^2)]*(3*(a/rh^2) - B^2/rh^2 - 1) + 1); 
A[r_] = -(a/r^2); 
p[r_] = ((9*a - B^2)*Log[Sqrt[6*a^2 - B^4]*Sqrt[(6*a^2 - B^4)/r^2 + 9*a - B^2] + 
        6*(a^2/r) - B^4/r])/Sqrt[6*a^2 - B^4] + 
    (1/r)*Sqrt[(6*a^2 - B^4)/r^2 + 9*a - B^2] - 
    ((9*a - B^2)*Log[Sqrt[9*a - B^2]*Sqrt[6*a^2 - B^4]])/Sqrt[6*a^2 - B^4]; 
As[r_] = A[r] + Sqrt[1/6]*p[r]; 
h[r_] = 1; 
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  • $\begingroup$ Why not to use NDSolve? $\endgroup$ Oct 23, 2023 at 11:16
  • $\begingroup$ @AlexTrounev My variable "l" takes all the real numbers and when i use NDSolve its giving me an error "non-numerical value at l==0". $\endgroup$
    – justJasper
    Oct 23, 2023 at 11:35
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    $\begingroup$ If a solution is found, what will you do next with it? $\endgroup$ Oct 23, 2023 at 14:00
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    $\begingroup$ It is highly unlikely that this problem has a symbolic solution. To obtain a numerical solution, you must specify the values of all constants, e.g., r0, rh, a, and B. $\endgroup$
    – bbgodfrey
    Oct 23, 2023 at 17:27
  • $\begingroup$ @AlexTrounev I am trying to find value of some coefficients which depends on this r[l] and further to determine eigen values of a Sturm-Liouville equation. $\endgroup$
    – justJasper
    Oct 24, 2023 at 8:28

1 Answer 1

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With no information about the metric tensor, the separation of variables yields

      Solve[dr/dl == Sqrt[g11[r]*gtt[r] - g11[r0]*gtt[r0]]/
                        Sqrt[grr[r]*gtt[r]*g11[r]], dl]

    dl -> (dr Sqrt[g11[r] grr[r] gtt[r]])/
               Sqrt[ g11[r] gtt[r] - g11[r0] gtt[r0]]

The integral cannot be solved, of course, symbolically

$$ l-l_0 = \int \frac{\sqrt{\text{g11}(r)\ \text{grr}(r) \ \text{gtt}(r)}}{\sqrt{\text{g11}(r)\ \text{gtt}(r)-\text{g11}(\text{r0}) \ \text{gtt}(\text{r0})}} \, dr$$

Insert functions and get the inverse

   InverseFunction[ Function[r,integral[r] ]][l-l_0]
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