[Due date 2020/6/26] Modern Differential Equations : Solution Essays

(24 points) An object with mass 4 kg is attached to both a spring with spring constant 1 N/m and a dash-pot with damping constant γ (measured in units Ns/m). (a) (2 points) Let y(t) be the displacement of the object from equilibrium at time t. Write a second order linear differential equation for y(t) that describes this system. (b) (3 points) Convert the differential equation from part (a) into an equivalent first order system of differential equations. (c) (4 points) For what positive value of γ does the system in part (b) have repeated eigenvalues? Call this value γ0. When γ = γ0, is the mass-spring system overdamped, underdamped, or critically damped? (d) (12 points) When γ = γ0, find the general solution of the first order system from part (b). (e) (3 points) When γ = γ0, sketch a phase portrait for the first order system from part (b).