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]

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}]