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][1]][1]

Since the values are so small, Using [`Chop`](http://reference.wolfram.com/language/ref/Chop.html) returns zero ("Chop uses a default tolerance of 10^-10").


  [1]: https://i.sstatic.net/QjRM1.png