Consider the system of differential equations :91 = 03:1 + 2a:2 2:"2 = 102:1 + 81:2 Our goal is first to find the general solution of this system and then a particular solution. In all your answers below, use the scientific calculator notation. For instance 3 + 52' is written 3 + 5*i and Ste3t is written 5*t*eA(-3*t}. a) This system can be written using matrices as X I = AX, where X is in R2 and the matrix A is A: ab sin(a) 1\" co a :2 till 0 0 2 10 8 b) Find the eigenvalue /\\ of the matrix A with the positive imaginary part and an eigenvector V associated to it. -a V: c) The general solution of the system of differential equations is of the form X: CIXI +62X2, where cl and c2 are constants, and X1 and X2 are the real and imaginary parts of a complex solution. [X 1 X2] = ab sin(a) 3 f so a a mi 0 2e4 tcos (2 t) e4 llsin (2 t) e4 1003(2 t) +2 e4 tsin (2 t B 5e4 recs (2 t) 534 tsin (2 t) [X 1 X2] denotes a matrix with columns X1 and X2 respectively. d) Find the solution if the initial condition is ($1) = ( 32) at t = 0. Answer: X(t) = (28) = 2 Use the scientific calculator notation to define the components 11:1 (13) and $2 (15). For instance 5te_3t is written 5*t*eA(-3*t)