Nonlinear first order equation with DSolve

Question

I am trying to solve a geodesic equation on Mathematica and would like to get a precise result, using DSolve would be a good start for me. My problem is the following: I have a squared expression for my DE and manage to get a solution with NDSolve for my equation using the Method -> {"EquationSimplification" -> "Residual"} option but cannot find anything similar using DSolve. Is there a way to ask Mathematica to find an analytical solution for this? I think the problem comes from the fact that two solutions are possible with this equation: the one with r[x] increasing or the one with r[x] decreasing and then increasing again, I have solved this with the "r'[0] == -10" term with NDSolve.

Thanks in advance!

Here is the code:

eqr = 0 == -(r'[x])^2 + En^2 - L^2*(1 - 2 mass/r[x])/r[x]^2
Rxn = NDSolve[{eqr, r[0] == 10000, r'[0] == -10, r'[0] == -10} /. {En -> 10, L -> 100, 
     mass -> 1}, r, {x, 0, 10000}, 
   Method -> {"EquationSimplification" -> "Residual"}][[1, 1, 2]]
Rxa = DSolve[{eqr, r[0] == 10000} /. {En -> 10, L -> 100, mass -> 1}, r, 
  x]

The DSolve returns the error "DSolve::bvimp: General solution contains implicit solutions. In the boundary value problem, these solutions will be ignored, so some of the solutions will be lost"

Answer 1

There is no explicit analytical solution only an implicit which can't be solved analytically for $r(x)$.

Clear["Global`*"];
ode = -(r'[x])^2 + En^2 - L^2*(1 - 2 mass/r[x])/r[x]^2 == 0

First@DSolve[ode /. {En -> 10, L -> 100, mass -> 1}, r[x], x]

And Mathematica can't solve this for $r(x)$. Giving one BC shows this

 DSolve[{ode, r[0] == 10000}, r[x], x]
 (* {} *)

DSolve::bvimp: General solution contains implicit solutions. In the boundary value problem, these solutions will be ignored, so some of the solutions will be lost.

Maple also gives an implicit solution, which is simpler, still the integral it gives can't be solved analytically

restart;
ode:= En^2- L^2*(1- (2*mass)/r(x))/r(x)^2 - diff(r(x),x)^2=0;
En:=10; L:=100; mass:=1;
ic:=r(0)=10000,D[1](r)(0)=-10;
sol:=dsolve([ode,ic],r(x));
DEtools:-remove_RootOf(sol)

$$ -x+\int_{{\it \_b}}^{r \left( x \right) }\!-{\frac {{{\it \_a}}^{2}}{ 10}{\frac {1}{\sqrt {{\it \_a}\, \left( {{\it \_a}}^{3}-100\,{\it \_a} +200 \right) }}}}\,{\rm d}{\it \_a}+{\it \_C1}=0 $$

Notice the upper limit is the solution. So it is implicit.

  Integrate[ - a^2/(10 Sqrt[a (a^3 - 100 a + 200)]), a] // InputForm

This is helpless to try to continue along this route.

Tried

AsymptoticDSolveValue[{ode /. {En -> 10, L -> 100, mass -> 1}, 
  r[0] == 10000, r'[0] == -10}, r[x], {x, 0, 4}]

But it could not solve it even using Asymptotic methods.

I would suggest to stick to numerical solver for such complicated nonlinear ODE's. May be someone else have other ideas.