(a) Find values of k₁ < 0 for which the system in Problem 11 is oscillatory.

(b) Consider the initial-value problem

x'' = x + k₁x³ = , x(0) = 0, x'(0) = 0.

Find values for k₁ < 0 for which the system is oscillatory.

Problem 11

The model mx'' + kx + k₁x³ = F₀cos ωt of an undamped periodically driven spring/mass system is called Duffing s differential equation. Consider the initial-value problem x'' + x + k₁x³ = 5 cos t, x(0) = 1, x'(0) = 0. Use a numerical solver to investigate the behavior of the system for values of k₁ > 0 ranging from k₁ = 0.01 to k₁ = 100. State your conclusions.