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Problems with NDSolve and initial conditions (NDSolve::nbnum1:)

Mathematica Asked by Matías López on June 17, 2021

this is the situation, I’m trying to solve a system (s1) with two differential equations, called "x[Ne]" and "y[Ne]". They have and initial condition that first need to satisfy another NDSolve for "phi[NN]". Since I put this as initial condition, problemas like NDSolve::nbnum1:, NDSolve::ecboo:, appears, and I honestly don’t understand why. Before I put this initial condition starting from "phi[NN]", the system for "x[Ne]" and "y[Ne]" has initial condition "x[0]==x0, y[0]==y0", where "x0" and "y0" was free parameters and no solutions coming from the differential equation for "phi[NN]", and I didn’t get any trouble. Any kind of idea with these situations? I read about "reap" and "sow" but i didn’t understand how to stick with my code:

"condition for x0";
Sol[Phi]f[[Delta]_, [Xi]_, n_] := 
 FindRoot[([Phi]f^(-2 - n/
     2) (-1 + Sqrt[
      1 + 4 [Delta] [Phi]f^(-1 + 
         n) (-4 + n + 8/(2 + [Xi] [Phi]f^2))]))/(2 [Delta]) == 
   1, {[Phi]f, 1}, WorkingPrecision -> MachinePrecision]
[Phi]f[[Delta]_, [Xi]_, n_] := [Phi]f /. 
  Sol[Phi]f[[Delta], [Xi], n]

sr[[Delta]_, [Xi]_, n_] := 
 NDSolve[{Derivative[1][[Phi]][
     NN] == ([Phi][
      NN]^-n (2 + [Xi] *[Phi][NN]^2) (-1 + Sqrt[
       1 + 4 [Delta]*[Phi][NN]^(-1 + 
          n) *(-4 + n + 8/(2 + [Xi] *[Phi][NN]^2))]))/(
    4 [Delta]), [Phi][0] == [Phi]f[[Delta], [Xi], 
     n]}, [Phi], {NN, 0, 70}, WorkingPrecision -> MachinePrecision]

xCMBsr[[Delta]_, [Xi]_, NN_] := [Phi][NN] /. sr[[Delta], [Xi], 2]
Vo[[Delta]_, [Xi]_, NN_, 
  n_, [Lambda]_] := ([Pi]^2 [Lambda] xCMBsr[[Delta], [Xi], 
     NN]^(-3 n) (2 + [Xi] xCMBsr[[Delta], [Xi], NN]^2)^3 (-1 + 
      Sqrt[1 + 
       4 [Delta] xCMBsr[[Delta], [Xi], NN]^(-1 + 
         n) (-4 + n + 8/(
          2 + [Xi] xCMBsr[[Delta], [Xi], NN]^2))])^2 (1 + 
      2 Sqrt[1 + 
        4 [Delta] xCMBsr[[Delta], [Xi], NN]^(-1 + 
          n) (-4 + n + 8/(
           2 + [Xi] xCMBsr[[Delta], [Xi], NN]^2))])^(3/2))/(8 Sqrt[
    3] [Delta]^2 (1 + 
      4 [Delta] xCMBsr[[Delta], [Xi], NN]^(-1 + 
        n) (-4 + n + 8/(2 + [Xi] xCMBsr[[Delta], [Xi], NN]^2)))^(
    1/4))
"condition for y0";
y0[[Delta]_, [Xi]_, NN_, n_, [Lambda]_] := 
  PowerExpand[
    1/(2 Sqrt[6] [Delta]) xCMBsr[[Delta], [Xi], NN]^(-n/2) Sqrt[
     2 + [Xi] xCMBsr[[Delta], [Xi], NN]^2]*Sqrt[
     Vo[[Delta], [Xi], NN, 
      n, [Lambda]]] (1 - 
       1 Sqrt[1 + 
         4 [Delta] xCMBsr[[Delta], [Xi], NN]^(-1 + 
           n) (-4 + n + 8/(
            2 + [Xi] xCMBsr[[Delta], [Xi], NN]^2))])] // 
   FullSimplify;
[Alpha][[Delta]_, [Xi]_, NN_, n_, [Lambda]_] :=  ([Delta]^(1/3))/
  Vo[[Delta], [Xi], NN, n, [Lambda]]^(1/3);
"system i want to solve";
s1[[Delta]_, [Xi]_, NN_, n_, [Lambda]_] := 
 NDSolve[{Derivative[1][x][
     Ne] == -((3 (2 + [Xi] x[Ne]^2) y[Ne])/(3 [Xi] x[Ne] y[Ne] + 
         3 [Alpha][[Delta], [Xi], NN, n, [Lambda]] y[Ne]^3 - 
         Sqrt[3] [Sqrt]((2 + [Xi] x[Ne]^2) (2 Vo[[Delta], [Xi], 
                  NN, n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                  n, [Lambda]]^2 x[Ne]^2 + y[Ne]^2) + 
             3 ([Xi] x[Ne] y[Ne] + [Alpha][[Delta], [Xi], NN, 
                  n, [Lambda]] y[Ne]^3)^2))), 
   Derivative[1][y][
     Ne] == -((3 (2 Vo[[Delta], [Xi], NN, 
             n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
             n, [Lambda]]^2 x[Ne] (2 + [Xi] x[Ne]^2)^2 + 
           3 y[Ne] ([Xi] x[Ne] + [Alpha][[Delta], [Xi], NN, 
                n, [Lambda]] y[Ne]^2)^2 (-3 [Xi] x[Ne] y[Ne] - 
              3 [Alpha][[Delta], [Xi], NN, n, [Lambda]] y[Ne]^3 + 
              Sqrt[3] [Sqrt]((2 + [Xi] x[
                    Ne]^2) (2 Vo[[Delta], [Xi], NN, 
                    n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                    n, [Lambda]]^2 x[Ne]^2 + y[Ne]^2) + 
                  3 ([Xi] x[Ne] y[Ne] + [Alpha][[Delta], [Xi], NN,
                     n, [Lambda]] y[Ne]^3)^2)) + 
           2 (1 + 1/
               2 [Xi] x[Ne]^2) (-4 Vo[[Delta], [Xi], NN, 
                n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                n, [Lambda]]^2 [Xi] x[Ne]^3 + 
              x[Ne] ([Xi] (-2 + 3 [Xi]) - 
                 6 Vo[[Delta], [Xi], NN, 
                   n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                   n, [Lambda]]^3 x[Ne]) y[Ne]^2 + 
              3 [Alpha][[Delta], [Xi], NN, 
                n, [Lambda]] (-1 + [Xi]) y[Ne]^4 + 
              Sqrt[3] y[
                Ne] [Sqrt]((2 + [Xi] x[Ne]^2) (2 Vo[[Delta], [Xi],
                     NN, n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                    n, [Lambda]]^2 x[Ne]^2 + y[Ne]^2) + 
                  3 ([Xi] x[Ne] y[Ne] + [Alpha][[Delta], [Xi], NN,
                     n, [Lambda]] y[Ne]^3)^2))))/((-3 [Xi] x[Ne] y[
             Ne] - 3 [Alpha][[Delta], [Xi], NN, n, [Lambda]] y[
             Ne]^3 + 
           Sqrt[3] [Sqrt]((2 + [Xi] x[Ne]^2) (2 Vo[[Delta], [Xi], 
                    NN, n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                    n, [Lambda]]^2 x[Ne]^2 + y[Ne]^2) + 
               3 ([Xi] x[Ne] y[Ne] + [Alpha][[Delta], [Xi], NN, 
                    n, [Lambda]] y[Ne]^3)^2)) (2 + [Xi] x[Ne]^2 + 
           3 ([Xi] x[Ne] + [Alpha][[Delta], [Xi], NN, 
                n, [Lambda]] y[Ne]^2) ([Xi] x[Ne] + 
              
              3 [Alpha][[Delta], [Xi], NN, n, [Lambda]] y[
                Ne]^2) - 
           2 Sqrt[3] [Alpha][[Delta], [Xi], NN, n, [Lambda]] y[
             Ne] [Sqrt]((2 + [Xi] x[Ne]^2) (2 Vo[[Delta], [Xi], 
                    NN, n, [Lambda]] [Alpha][[Delta], [Xi], NN, 
                    n, [Lambda]]^2 x[Ne]^2 + y[Ne]^2) + 
               3 ([Xi] x[Ne] y[Ne] + [Alpha][[Delta], [Xi], NN, 
                    n, [Lambda]] y[Ne]^3)^2)))), 
   x[0] == xCMBsr[[Delta], [Xi], NN], 
   y[0] == y0[[Delta], [Xi], NN, n, [Lambda]]}, {x, y}, {Ne, 0, 
   70}, WorkingPrecision -> MachinePrecision]

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