Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The matrix of eigenvectors and its inverse is 1 0 1 S 0 -1 0 0 1 1 -1 -1 -1 S 1 1 0 0 1 0 1 The Jordan canonical form of the matrix which satisfies J S 1AS is 2 J 0 0 1 2 0 0 0 3 Recall that the function of a Jordan block is A 1 0 0 f A f A 1 f A 2 f A A 3 f 0 A 1 0 0 f a f A 1 f A 2 0 0 A 1 0 0 f A f A 1 I 0 0 0 A J 0 0 0 f A and that the function of a matrix in Jordan canonical form is J1 0 0 0 f J1 0 0 0 0 J2 0 0 0 f J2 0 0 0 0 J3 0 0 0 f J3 0 U0 0 0 J4ỳ J 0 0 0 f J4 7 We want to compute eJt so we consider the function f A we see that e2i t e2t eJt I 0 e2t 00 eAt which has the derivative f A teAt. Thus 0 0 e3t 894 The exponential matrix is eAt S eJt S-1 e2t - 1 t e2t e3t e2t e3t eAt 0 e2t 0 0 e2t e3t e3t The general solution of the homogeneous differential equation is _ A t x eAt c. 2. The solution of the inhomogeneous differential equation subject to the initial condition is x eAt 0 eAt i e Ar g T dT Jo x eAt r A g T dT o Solution 1. dx dt t x a b c d I The first component of this equation is tx 1 ax1 bx2. .
