TC2411 Tutorial 5 1. Determine the eigenvalues and eigenvectors for 1 2 1 1 0 1 4 4 5 2. Show by direct computation that for the symmetric matrix 1 1 2 1 3 0 2 0 2 The eigenvalues are real, and the eigenvectors form orthogonal set of vectors. dx dy = x + 2y = 4x + 3y dt dt dx dy = 2x + 4 y = x + 6 y with x (0) = 1 y (0) = 6 4. dt dt dx dy = x + y with x (0) = 2 y (0) = 8 = x+ y 5. dt dt dy1 dy 2 = 4 y1 + 4 y 2 + 12 , = 1.6 y1 + 1.2 y 2 + 4.8 6. dt dt 3. with y1 and y2 being zero initially. 7. dy1 = 2 y1 + y2 4 , dt dy2 = 4 y1 3 y2 + 7 dt with y1 = 1 and y2 = 2 at t = 0 8. Consider 1st order chemical reactions: A B C . The forward and reverse rate constants for both reactions are all 1 s-1. The initial concentration of A is 1 M and the other two species are absent. Determine the concentration of each species as a function of time. dy1 dy2 = y1 y2 + 1 = y1 2 y2 + t , 9. dt dt with y1 = 1 and y2 = 0 at t = 0 Answer 1 2 1 1. eigenvalues 1,2 and 3, eigenvectors: 1 , 1 , 1 2 4 4 1 2 3 2 + 3 2. eigenvalues 4, 1 3 , eigenvectors: 1 , 1 , 1 1 3 1 1 3 x 1 1 3. = c1 e 5t + c2 e t y 2 1 2t + 1 4t 2 x 4. = 7 e 4t + 13 e t +1 1 y cos t t sin t t x 5. = 2 e + 8 e sin t cos t y 2 t 0.8 t y 2 = 4e 2 t + 4e 0.8t +3 6. y1 = 8e + 5e y 1 1 5 / 2 1 7. 1 = e 2 t e t + 1 1 y2 2 4 yA 1 1 1 1 1 t 1 3t 8. y B = 1 0 e + 2 e 3 2 6 1 1 yC 1 9. y1 = 2 cos t + t 1 , y 2 = cos t + sin t + t 1