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I need help to find a solution to the differential equation bellow. The solution obtained with this code is a very small negative number in all the domain, and it was mathematically proved that exists a positive solution to this problem. Any hints?

solu = NDSolve[{-z''[t] == (Sin[2 + z[t]] z[t]^2)/(1 + t)^4.5,z[0] == 0., z'[10000] == 0.}, z, {t, 0., 10000}, Method -> {"Shooting","StartingInitialConditions" -> {z[0] == 0., z'[0] == 0.15}}]
LogLinearPlot[Chop[Evaluate[z[t] /. solu]], {t, 0.1, 10000}]
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  • $\begingroup$ Well, the zero function is a solution that satisfies the ODE & BCS. (It's also, trivially, a polynomial.) $\endgroup$
    – Michael E2
    Commented Jul 9, 2016 at 3:59
  • $\begingroup$ I cannot see how you can expect to get non-trivial polynomial solutions... $\endgroup$
    – Per Alexandersson
    Commented Jul 9, 2016 at 4:16
  • $\begingroup$ It seems likely that this nonlinear ODE system has no solution except z == 0 due to the Sin[2 + z[t]] term in the ODE. Remove it, and solutions can be obtained easily, as in your earlier question. $\endgroup$
    – bbgodfrey
    Commented Jul 9, 2016 at 5:09
  • $\begingroup$ It was mathematically proved that exists a positive solution to this equation...but what we are obtaining is negative, i dont know why. $\endgroup$
    – Stratus
    Commented Jul 9, 2016 at 18:59
  • $\begingroup$ Note that Bob Hanlon's solutions are essentially zero. The negative values may be due only to rounding error. Do you get different negative solutions? $\endgroup$
    – Michael E2
    Commented Jul 10, 2016 at 5:17

1 Answer 1

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solu = NDSolve[{-z''[t] == (Sin[2 + z[t]] z[t]^2)/(1 + t)^(9/2), z[0] == 0, 
     z'[10000] == 0}, z, {t, 0, 10000}, 
    Method -> {"Shooting", 
      "StartingInitialConditions" -> {z[0] == 0, z'[0] == 15/100}}][[1]];

LogLinearPlot[z[t] /. solu, {t, 0.1, 10000}]

enter image description here

Since the values are so small, Using Chop returns zero ("Chop uses a default tolerance of 10^-10").

With higher precision the values are even smaller

solu = NDSolve[{-z''[t] == (Sin[2 + z[t]] z[t]^2)/(1 + t)^(9/2), z[0] == 0, 
     z'[10000] == 0}, z, {t, 0, 10000}, 
    Method -> {"Shooting", 
      "StartingInitialConditions" -> {z[0] == 0, z'[0] == 15/100}},
    WorkingPrecision -> 20][[1]];

LogLinearPlot[z[t] /. solu, {t, 1/10, 10000},
 WorkingPrecision -> 20]

enter image description here

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