(5 points) Write your SID: odd = 1 1- (30 points) Solve the following ordinary differential equation and find x(t) a. by direct integration. b. by LaPlace method. Explain which LaPlace transform from the LaPlace tables you are using. SID x2t = 0, x(0) = 1 1 x x - 2+2=0 X (0) =) 2- (20 points) Simplify to have a simple complex number of the form x + yj where x and y are real, j = i = 1: SID a. j200 b. j) 10 *) +\/200 3- (20 points) Linearize the nonlinear function f(x) = x + 1 around x = 1. 4- (30 points) A dynamic system is expressed by the following ODE where x(t) is the state and output of interest ; and the f(t) is the input. Initial condition(s) is/are zero. 2x+4x+2x= SID f(t) 2x + 4x + 2x +4x+2x= What is the characteristics equation of this dynamic system? a. What is the order of the system? b. C. What is (are) the pole(s)? d. Find the transfer function. e. f. 1 x f(+) If applicable, compute 3, T, wn, and wa. If not applicable, state the reason why. Find the state in Laplace domain X(s) if input f(t) is a unit impulse function.