Given the dynamic system x˙ = 2x + 3y y˙ = 3x + 2y

(i) Show that the characteristic roots of the system are r = 5 and s = −1.
(ii) Derive the eigenvectors associated with the eigenvalues obtained in (i).
(iii) Show that the solution values are:
x(t) = c1e5t + c2e−t y(t) = c1e5t − c2e−t and verify that c1e5t 
1 1 
, c2e−t  1 −1 
are linearly independent.
(iv) Given x(0) = 1 and y(0) = 0, show that x(t) = 1 2 e5t + 1 2 e−t y(t) = 1 2 e5t − 1 2 e−t