Differential equation and linear algebra.

1. (Mass-spring with variable damping) The figure below depicts a typical mass-spring dashpot system where m denotes the mass of the oscillating object (in kilograms), k denotes the spring constant (in Newtons/meter), and c denotes the damping constant. We use (t) to denote the position of the mass at time t (with x 0 being its equilibrium position); positive directions of x imply the mass is to the right of its equilibrium position. Throughout, we fix the values m-3 and k = 48, However, the value of the damping constant c will vary. In this setup, x(t) satisfies the ODE: 3r"(t) cr'(t) +48x(t)0 which after dividing through by the mass 3 is equal to +16(t) 0 Throughout this problem, "IVP" will refer to this ODE subject to the initial conditions x(0)1 and (-2 (a) Solve the IVP when there s no damping, ie, when c-0. After finding your solution in linear combination form, convert into amplitude-phase form of C cos(wot - a). Identify the num erical values for amplitude C, phase angle ?, and time lag and the mass does not oscillate critical damping whenever the associated characteristic polynomial has a double real root (b) Over-damped case: Solve the IVP when c 30. In this case, the dashpot is very strong (c) Critically damped case: Recall that a value of the damping constant induces so-called Confirm that c 24 leads to critical damping, and solve the IVP in this case polynomial to three digits. display containing the graphs of all four solutions above, on the interval 0t