1. Consider the system described by the following first order differential equation y = -2y + 10u. (a) (2pt) Using the "ad hoc" approach (generate a particular solution first and then add a homogeneous solution with a "free" parameter a such that the total solution satisfies the given initial condition), compute the total solution for u(t) = et for t 2 0, y(0) = 1. (b) (2pt) Using the "ad-hoc" approach, compute the total solution for u(t) = sint for t 2 0, y(0) = -1. 2. Consider the following first order system y (t ) = ay(t) + bu(t ) y(0-) = 0 u (t ) = cate-ctu(t ) where a 0 and u(t) is a unit step function. (a) (2pt) Show that Lou(r)ar = 1. (b) (2pt) Compute the system response using y ( t ) = eaty ( 0- ) + ( ealt - 7 ) bu ( T ) dt . (c) (2pt) Show that the system response (for a fixed a) becomes y ( t ) = eatb as c - oo, e.g., as the input becomes sharper and narrower, the system response becomes the ideal impulse response. (d) (5pt) For the following first order system y (t ) = -2y(t) + 10u(t) y(0) = 0 u(t) = cate-ctu (t ) plot the ideal impulse response as well as the system responses for c = 1, c = 10 and c = 1000 in the same figure using different colors. Define the time vector as t = [0:0.001:5]