Question

Hi, I need the full worked solution/explanations for all parts of this questions please. The final answers to each part are shown below the question. Clear handwriting is greatly appreciated. Thank you! :)

Question 5 (a) Solve the eigenvalues and its corresponding eigenvectors of a 2x2 matrix given by 2 0 (8 marks) For the system of differential equations, Зх — у ў 2х + 6е ". (b) Write and explain the system of differential equations in matrix form. (2 marks) Find the complementary function (c) (3 marks) (d) Calculate the solutions of the system of differential equations which satisfy the initial conditions, x(0) = 0 and y(0) 0. (12 marks) Final Answers: Eigenvalues are 1 and 2 with eigenvectors are [1 2]T and [1 1]T respectively (a) 5. 3 -1 0 (b) | 2 0 Хс = A |yc] eB e2 (c) = 3 e - 2 (d) e2 _

Answer 1

for eigenvalues, 137 +1=0 12 ord » d ²-3x+2=0 (4-1)(x-2)=0 d=1,2 a Eigenvector of 1 > 2x y = 0 4 y = 2x ie (nry 1= x (1,25 2x-y so I % eigen vector : [1, 2] Eigen vector of 2 (3-2 tiralo 12 0-2 Lyo - x - y = 0 x=y ie (wigs? = -x (111) 20-2y=0 . Eigenvetor : (1,1) (6 À 32-y | 0 = 2x +6e-t (6) complementary funtion dit c) - Alvi + Benet 1906 A +80*11

a Ca) (19)+ Bem Then particular integral, up) = X J x Gotldt - (et ezt Ir (Gert bet het ezt) | +6e3t) . Sution G) - Ael(!) + Be*(1) -et (4) 200)=0 14co)=0 → * )= A(12)+B (1) - W A + B = 0 2A +B-4=0 ; B=2