Mathematical Methods and Mechanics

Question 5

(a) Forthefollowing3 x 3mafiix

lz o 1lttl-r 4 -rl l-, z ,J

(i) calculate the three eigenvalues, and (9 marks)

(ii) calculate the three corresponding eigenvectors. (10 marks)

(b) The eigenvalues of a certain 2 x 2 matrix B are 1 and -2. Let

A=2B'3+88-1-7I

Calculate the values of the trace and the determinant of A. Show how you obained youranswers

(6 marks)

Question 6

The question concerns the system of differcntial equations

I *= y-3 lj,=*r-ry+2

(a) Write the system in the form

L*,i7r : V(x"y) where V(x, y) is a vector field.

(2 marks)

O) Calculate all the equilibrium points forthe system of differential equations. (6 marks)

(c) For each of the equilibrium points (lto, Io) define the excesses a and v by x= uaXo and y=v + Is andusetheseto finda linearapproximation t = M u forthe system of equations nearthe equilibrium point.

(8 marks)