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I have a second order differential equation which is really Long and It is second derivative of y with respect to x with 4 other parameters , E- energy, L-angular momentum, delta that takes a few positive integer values and epsilon that can be either 0 or 1. how should I integrate such a thing in Wolfram mathamtica? I use DSolve & after a few minutes I get the equation I've written back don't know this means it's not solvable analytically or there is a mistake somewhere. A1d is the equation that needs to be solved. I've also included what I've done in mathematica.

enter image description here

enter image description here

here is the code

    DSolve[(E^2 - L^2*((x - 1)^(2*d - 1)/((1 - y[x]^2)*(x + 1)^(2*d + 1))) - a*((x - 1)/(x + 1))^d)*Derivative[2][y][x] == 
   (E^2 - L^2*((x - 1)^(2*d - 1)/((1 - y[x]^2)*(x + 1)^(2*d + 1))) - a*((x - 1)/(x + 1))^d)*(((d^2 - 1)*y[x]*(1 - y[x]^2))/((x^2 - 1)*(x^2 - y[x]^2))) - 
    L^2*((y[x]*(x - 1)^(2*d - 2))/((1 - y[x]^2)*(x + 1)^(2*d + 2))) + 
    (a*((x - 1)^(d - 1)/(x + 1)^(d + 1)) - (E^2 - L^2*((x - 1)^(2*d - 1)/((1 - y[x]^2)*(x + 1)^(2*d + 1))) - a*((x - 1)/(x + 1))^d)*
       ((2*(x - d)*(x^2 - y[x]^2) + (d^2 - 1)*x*(1 - y[x]^2))/((x^2 - 1)*(x^2 - y[x]^2))) - L^2*(((x - 2*d)*(x - 1)^(2*d - 2))/((1 - y[x]^2)*(x + 1)^(2*d + 2))))*
     Derivative[1][y][x] - ((E^2 - L^2*((x - 1)^(2*d - 1)/((1 - y[x]^2)*(x + 1)^(2*d + 1))) - a*((x - 1)/(x + 1))^d)*
       (y[x]*((x^2 - y[x]^2 + (1 - d^2)*(1 - y[x]^2))/((1 - y[x]^2)*(x^2 - y[x]^2)))) + L^2*((y[x]*(x - 1)^(2*d - 1))/((1 - y[x]^2)^2*(x + 1)^(2*d + 1))))*
     Derivative[1][y][x]^2 + (((d*a)/(1 - y[x]^2))*((x - 1)/(x + 1))^d - (E^2 - L^2*((x - 1)^(2*d - 1)/((1 - y[x]^2)*(x + 1)^(2*d + 1))) - a*((x - 1)/(x + 1))^d)*
       ((x*(x^2 - 1) + d^2*x*(1 - y[x]^2) - 2*d*(x^2 - y[x]^2))/((1 - y[x]^2)*(x^2 - y[x]^2))) - L^2*(((x - 2*d)*(x - 1)^(2*d - 1))/((1 - y[x]^2)^2*(x + 1)^(2*d + 1))))*
     Derivative[1][y][x]^3, y[x], x]

and the link to the article is this arxiv.org/pdf/gr-qc/0304064.pdf

the equations are almost at the end of the article.

Danny Nelson
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    Can you at least point to a reference where your DE is described, if you are unable to type out the equation here? – J. M.'s missing motivation Apr 04 '17 at 09:10
  • @J.M. thank you for taking the time to answer. I've included screen shots. – Danny Nelson Apr 04 '17 at 09:25
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    The screenshot of the equation is okay, except that you forgot to mention where (book, paper, etc.) you got it from. The screenshot of your Mathematica code is not; here, we need the code to be something we can copy and paste into our own Mathematica sessions. – J. M.'s missing motivation Apr 04 '17 at 09:36
  • @j.M. https://www.scribd.com/document/343991561/EQ here is a text file of the code – Danny Nelson Apr 04 '17 at 10:52
  • You might want to look up E and this advice (#4 in particular). – Michael E2 Apr 04 '17 at 11:16
  • @MichaelE2 thank you so much, figured it out how to post the code here. hope somebody sees how to solve this now – Danny Nelson Apr 04 '17 at 11:21
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    Thanks for updating. Such a complicated nonlinear ODE might not be solvable by DSolve, but E in Mathematica is the base of the natural log., 2.71828.... Maybe someone will figure it out. – Michael E2 Apr 04 '17 at 11:26
  • @MichaelE2 that is what I'd really like to know. correct me if I'm wrong DSolve means solve analytically? if so is there a way to understand if this is not solvable analytically? perhaps mathematics shows a message? PS. I will substitute E with some other letter now to see if it works, thank you for helping. – Danny Nelson Apr 04 '17 at 11:28
  • @MichaelE2 substituting E with p gives the equation back again, and no solution as before – Danny Nelson Apr 04 '17 at 11:32
  • Returning the original code indicates that DSolve could not figure out how to solve it. (I didn't think changing E would fix it, because it seems like a reasonable energy level. It was just to point out it has a specific meaning.) Sometimes an ODE is still solvable after a transformation (human insight). Numerical solvers are NDSolve (must give numeric values to all parameters) and ParametricNDSolve (returns a function of parameters). You can plot them & functions of them. Your ODE has some singularities, so it might not be completely straightforward. – Michael E2 Apr 04 '17 at 12:10
  • Name your equation eq. Then, Collect[(Subtract @@ eq /. E -> 0 /. d -> 0 /. L -> 0)/a, {y''[x], y'[x]}, Simplify] == 0 greatly simplifies the equation, but DSolve still cannot handle it. Perhaps, some trigonometric substitution would help, although I am not optimistic. In any case, posting this simpler equation is more likely to elicit help. – bbgodfrey Apr 04 '17 at 13:20
  • @MichaelE2 I'd want to try moving to prolate spheroidal coordinates. where$$ \rho = \sigma \sqrt (x^2-1)(1-y^2) and z= \sigma xy $$ the equation is two big to do it by hand, can I do this by mathematica? – Danny Nelson Apr 04 '17 at 13:59
  • @bbgodfrey I tried it, I got 1/0 indeterminate. I think I should the do coordinate change then try to plot it numerically, can you help with this? – Danny Nelson Apr 04 '17 at 14:02
  • There's TransformedField and CoordinateTransformData; and Kuba's DChange. I don't have time to give it a really hard look, though. – Michael E2 Apr 04 '17 at 14:42

1 Answers1

0

I was able to run this.

Clear[y, d, L, a]
y'[x] =.
d = 1;
L = 2;
 a = 3;
NDSolve[{(E^2 - 
       L^2*((x - 1)^(2*d - 1)/((1 - (y[x]^2))*(x + 1)^(2*d + 1))) - 
       a*((x - 1)/(x + 1))^d)*
     y''[x] - (E^2 - 
       L^2*((x - 1)^(2*d - 1)/((1 - (y[x]^2))*(x + 1)^(2*d + 1))) - 
       a*((x - 1)/(x + 1))^d)*(((d^2 - 1)*
         y[x]*(1 - (y[x])^2))/((x^2 - 1)*(x^2 - (y[x]^2)))) - 
    L^2*((y[x]*(x - 1)^(2*d - 2))/((1 - (y[x]^2))*(x + 1)^(2*d + 
            2))) + (a*(((x - 1)^(d - 1))/((x + 1)^(d + 1))) - (E^2 - 
          L^2*((x - 1)^(2*d - 1)/((1 - (y[x]^2))*(x + 1)^(2*d + 1))) -
           a*((x - 1)/(x + 1))^
            d)*((2*(x - d)*(x^2 - (y[x]^2)) + (d^2 - 1)*
             x*(1 - (y[x]^2)))/((x^2 - 1)*(x^2 - (y[x]^2)))) - 
       L^2*(((x - 
              2*d)*(x - 1)^(2*d - 2))/((1 - (y[x]^2))*(x + 1)^(2*d + 
               2))))*(y'[
       x]) - ((E^2 - 
          L^2*((x - 1)^(2*d - 1)/((1 - (y[x]^2))*(x + 1)^(2*d + 1))) -
           a*((x - 1)/(x + 1))^d)*(y[
           x]*(x^2 - (y[x]^2) + (1 - 
                d^2)*(1 - (y[x]^2)))/((1 - (y[x]^2))*(x^2 - (y[
                  x]^2)))) + 
       L^2*((y[x]*(x - 1)^(2*d - 1))/((1 - (y[x]^2))^2*(x + 1)^(2*d + 
               1))))*(y'[
        x])^2 + (((d*a)/(1 - (y[x]^2)))*((x - 1)/(x + 1))^
         d - (E^2 - 
          L^2*((x - 1)^(2*d - 1)/((1 - (y[x]^2))*(x + 1)^(2*d + 1))) -
           a*((x - 1)/(x + 1))^d)*((x*(x^2 - 1) + 
            d^2*x*(1 - y[x]^2) - 
            2*d*(x^2 - y[x]^2))/((1 - y[x]^2)*(x^2 - y[x]^2))) - 
       L^2*(((x - 
              2*d)*(x - 1)^(2*d - 1))/((1 - y[x]^2)^2*(x + 1)^(2*d + 
               1))))*(y'[x])^3 == 0, y[0] == 2, y'[0] == 3}, 
 y[x], {x, 0, 30}]
Dr. user44690
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