11. Consider a frictionless mass-spring system (standard mass attached to a spring on a frictionless table). Suppose the restoring force of the

11. Consider a frictionless mass-spring system (standard mass attached to a spring on a frictionless table). Suppose the restoring force of the spring is not given by Hooke’s Law, but instead is of the form F = −(ky + epsilony^3). If epsilon= 0, the assumption amounts to Hooke’s Law, but in this problem we shall focus on epsilon does not = 0 (either positive or negative). If we have air resistance present with damping coecient γ, then the equation of motion (assuming the displacement is indicated by the variable y) becomes

my'' + γy' + ky + epsilony3 = 0.

Henceforth we shall normalize by assuming that m = k = 1 and γ = 0, and then take the initial conditions

y(0) = 1, y'(0) = 0.

(a) Plot the solution when epsilon = 0. What is the amplitude and period of the solution?

(b) Let epsilon = 0.1. Plot a numerical solution. Is the motion periodic? Estimate the amplitude and period.

(d) How do A and T depend on epsilon?

IN MATLAB