y[x] = 1 + p x + (4/3) (x^((3/2))) + b (x^2) + c (x^((5/2)));

Series[
      (((D[y[x], {x, 2}]))^2) - (x^(-1)) (((y[x]))^3), {x, 0, 3}]

NSolve[(x^2) - x - 18 == 0, x]