11. Solve the following system X = X + 3 4 subject to the initial condition X (0) = 5 13et - 8e2t - 9 Ans: X = 13et + 12e2t + 6 12. (January 2006) (a) Solve the initial value problem X' = X, with x1(0) = 1, x2(0) = x3(0) = 0, where the matrix -2 haseigenvectors corresponding to eigenval- ues -1, 0 and 2 respectively. (b) Let B be a 2 x 2 matrix with eigenvalues -1 and 4. Write down the eigenvalues of each of the following matrices. (i) B12, (ii) (B + 312)-1. Ans: (a) x1 = eat, 12 = -1+e-, x3 =1-e", (b) (i) 412 and 1, (ii) 1/7 and 1/2 13. (February 2006) (a) An object moves in the plane in such a way that its position (71, 12) at any time t satisfies the simultaneous differential equations Find the particular solutions of z, and x2 that satisfy the initial con- ditions r (0) = 2 and r2(0) = 0. Ans: (a) m1 = eff + eat, r2 = -2e- + 2eltAns: (a) m = eff test, x2 = -2eff + 2ed 14. (October 2006) (a) Let A be a 2 x 2 matrix with eigenvalues -1 and -2. Find (i) the determinant of a matrix B, where B = A. (ii) the eigenvalues of a matrix C, where C = A~ + 412. (b) By using the concepts of eigenvalues and eigenvectors, solve the following system of differential equations: dt = -4m1 + 212 = 2x1 - 412 4 Ans: (a) (i) 64 (ii) 3 and 7/2. (b) a = cle-2 + cze-6, x2 =