For the dynamic system x˙ = x + 3y y˙ = 5x + 3y Show:

(i) that the two eigenvalues are r = 6 and s = −2

(ii) that the two eigenvectors are vr =

 1 5/3



and vs =

 1

−1



(iii) and that the general solution satisfying x(0) = 1 and y(0) = 3 is x(t) = 3 2 e6t − 1 2 e−2t y(t) = 5 2 e6t + 1 2 e−2t