Nonlinear Systems & Linearization · Hard problem

For the system $x\,\prime = y$, $y\,\prime = -x - x^3$, the quantity $E = \tfrac12y^2 + \tfrac12x^2 + \tfrac14x^4$ is conserved. A trajectory starts at $(x, y) = (1, 0)$. Find $E$. Express as a common fraction.