Question

A dynamic system is described by a set of ordinary numbers

(20 marks total) Question 3 A dynamic system is described by a set of ordinary differential equations: 0.5x=0.05x +0.1y y 0.1x Answer the following 4 questions about this system (please use answer book for working but provide the final answers in the workbook): (a) The above system can be rewritten in matrix form as x Ax where x is the vector with solutions: x(t) y(t) X Write down the system matrix A for this system (with calculated values) (3 marks) (b) Calculate the eigenvalues of the above matrix (6 marks) (c) Calculate the eigenvectors of the system matrix (8 marks) (d) Write down the general solution for x(t) and y(t) (3 marks)

Answer 1

0.5x =-0.057+0-1 0.17 =0.1+0.2 O.1X O.2 O. EAJ O.2 0.1 r-0.1-8 O.2 LA-A J -A 0. -AFa1-A)-02).) CAXAto.1) O,02 to1-0.02 -0 1 A-0.11 101)4 W0.02) 0.11 0.8 2 2 2-0.1 A=0.2

0.1-(-0.2) 0,2 -C-0.2) 0.) U-N 0.1 0.1 0.2 0.11+0.2-0-) T. - efn vectoon 0.1-0. 0.2 0. ) 0.) 0. 2 7 -01 O 0.2Y +0.2Y 0 T digen vetom

-0.2t Cre n, n-vehox O./6 +Ge 2 0.16 +2e Ge 0.26 O.16 -2 Ge + e