need help with a-c

2. Newton's second law for the damped harmonic oscillator yields a!2 d (@+27E+w2)x=0 (1) with 7 > 0 (a) Write this equation as a system of rst order differential equations using matrix notation (i.e., 2311" = Mu) with wt\") (b) Assuming M has two eigenvectors 61 and 62 with eigenvalues A1 and A2, show that Alt eMt = S (90 6221:) 81 (3) where S = (31 62). Hint: use the trick given in lecture that M k = SDkS_1. (c) Using the fact that u(t) = eMtuo is the solution to Eq. 1, solve the initial value problem for 15(0) = (:60 vo)T for the following cases by writing $(t) in terms of real valued functions (i.e., without terms like c\"): i. 72 > (.02. ii. 72