e the phase-plane method to show that the solution to the nonlinear second-order differential equation x" + 2x - x² = 0 that satisfies x(0) = 1 and x'(0) = 0 is periodic. 1,2 dy dx dt = y. Then the differential equation can be solved by separating variables. It follows that the general solution is 2/₁² = C+ x³ = x² dx y , and since X 0) = (x(0), x'(0)) = (1, 0) then the particular solution is But for each x such that 1-√√3 < x < 1, the particular solution has 1 X lue(s) of y. Therefore X(t) is a periodic solution. = 2/² = (x − 1) (x² – 2x − 2) X correspondi