(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 ? (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 di?erential equation for y(t) that describes this system. (b) (3 points) Convert the di?erential equation from part (a) into an equivalent ?rst order system of di?erential equations. (c) (4 points) For what positive value of ? does the system in part (b) have repeated eigenvalues? Call this value ?0. When ? = ?0, is the mass-spring system overdamped, underdamped, or critically damped? (d) (12 points) When ? = ?0, ?nd the general solution of the ?rst order system from part (b). (e) (3 points) When ? = ?0, sketch a phase portrait for the ?rst order system from part (b).