2. Newton's second law for the damped harmonic oscillator yields \left(\frac{d^{2}}{d x^{2}}+2 \gamma \frac{d}{d x}+\omega^{2}\right) x=0 (a) Write this equation as a system of first order differential equations using matrix

notation \text { (i.e., } \left.\frac{d u}{d t}=M u\right) \text { with } u=\left(\begin{array}{l} x \\ v \end{array}\right) (b) Find the eigenvectors and their eigenvalues of M. (c) Let the matrix E = (e1 e2) where ei are the eigenvectors. ^^20Find^^20e^{Mt}E. (d) Using the fact the u(t) = e^Mt uo is the solution to Eq. 1 and that e^{M t} u_{o}=e^{M t} E E^{-1} u_{o} solve the initial value problem for u(0)=\left(x_{o} v_{o}\right)^{T} (e) Consider the case where y2 > w². Write down an expression for x(t) in terms of real exponentials. This corresponds to the overdamped solutions (f) Consider the case where y² < w². Write down an expression for x(t) in terms of sines and cosines. This corresponds to the underdamped solution (g) Consider the case where y² = w² . What is the problem with our approach?