Nonlinear Systems & Linearization · Hard problem

For $x' = y,\; y' = 4x - x^3$, the energy $E = \dfrac{y^2}{2} + \dfrac{x^4}{4} - 2x^2$ is conserved. A solution starts at rest ($y=0$) at $x=1$. It oscillates between $x=1$ and one other value of $x > 1$ where $y = 0$ again. Find that value.