Consider the system described by the following differential equation: d2y dt2 3 dy dt 2y u (1) In this problem, we are interested in the solution to the associated homogeneous equation, i.e. set uptq 0. (a) What is the characteristic equation? What is the general solution of the homogeneous equation? Find the transfer function. (b) Convert the differential equation to state variable form: 9x A x B u y C x (c) Find the characteristic polynomial and its roots associated with this state variable equation. (d) Find the eigenvectors (v1, v2, . . ., vn) associated with each root (s1, s2, . . ., sn), and write the general solution of the homogeneous state space equation using arbitrary constants (p1, p2, . . ., pn), in the form of: xptq p1 v1 es1t p2 v2 es2t . . . pn vn esnt (2) (e) Write the solution vector xptq with the arbitrary constants evaluated in terms of the initial state, in the form of: xptq M es1t 0 . . . 0 0 es2t . . . 0 ... ... . . . ... 0 0