Problem III (15 points) Consider the following 2nd order nonlinear equation 3 x" + 2x² + x = 0. 1. (5 points) Write the differential equation as a 2 × 2 Hamiltonian system of (x, y) and write down its energy (hamiltonian) function E(x, y). 2. (10 points) Find the equilibrium solutions to the Hamiltonian system you construct. If (x(t), y(t)) is on the trajectory E(x, y) c, for what values c can the solution be periodic? = = 9 3. (10 points) If x(t) is a solution with initial condition x(0) = 0, x'(0) = 2√3, does the limit lim+→+∞ x(t) exit? If yes, find the limit. If no, state the reason.