Chapter 3.6, Problem 18E

In Exercises $13-20,$ consider harmonic oscillators with mass m, spring constant k, and damping coefficient $b$ . For the values specified,
(a) write the second-order differential equation and the corresponding first-order system:
(b) find the eigenvalues and eigenvectors of the linear system;
(c) classify the oscillator (as underdamped, overdamped, critically damped, or undamped) and, when appropriate, give the natural period;
(d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and
(e) sketch the $y(t)$ - and $v(t)$ -graphs of the solution with the given initial condition.
18. $m=9,k=1,b=6,$ with initial conditions $y(0)=1,v(0)=1$