3.45 Use the Poincar-Bendixson theorem (Theorem 3.36) to show that the system X X Y - 2", y' x + y - y3 = has a periodic solution. (Hint: Show that the system has an invariant square.) 3.43 Show that the differential equation x" + [x2 + (x')2 1]x' + x = 0 has a nontrivial periodic solution. Theorem 3.36 (Poincar-Bendixson) Consider equation (3.45) for the case n = 2. If o(t, x) is a bounded orbit for t> 0 and W is its w-limit set, then either W is a cycle, or for each y EW, the w-limit set of t, y) is a set of one or more equilibrium points. 3.45 Use the Poincar-Bendixson theorem (Theorem 3.36) to show that the system X X Y - 2", y' x + y - y3 = has a periodic solution. (Hint: Show that the system has an invariant square.) 3.43 Show that the differential equation x" + [x2 + (x')2 1]x' + x = 0 has a nontrivial periodic solution. Theorem 3.36 (Poincar-Bendixson) Consider equation (3.45) for the case n = 2. If o(t, x) is a bounded orbit for t> 0 and W is its w-limit set, then either W is a cycle, or for each y EW, the w-limit set of t, y) is a set of one or more equilibrium points