Harmonic motion. An unforced, damped oscillator is modeled by the equationwhere m is the m...

Harmonic motion. An unforced, damped oscillator is modeled by the equation

where m is the mass, c is the damping constant, and k is the spring, constant. Write the equation as a system of first order ODEs and create a function ODE file that allows the passing of parameters m, c, and k. In each of the following cases compute the solution with initial conditions y(0) = 1, and y′(0) = 0 over the interval [0, 20], Prepare both a plot of y versus t and a phase plane plot of y′ versus y.

a) (No damping) m = 1, c = 0, and k = 16.

b) (Under damping) m = 1, c = 2, and k = 16.

c) (Critical damping) m = 1, c = 8, and k = 16.

d) (Over damping) m = 1, c = 10, and k = 16.