hw6-solutions

Consequently the eigenvalues of a are given by a11

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 = −2 and a2 = −3. Thus, cos(t) − 5 sin(t) . −2 cos(t) − 3 sin(t) y (t) = e−2t Problem 23: (a) The characteristic equation is 0 = det(λI − A) = det λ − a11 −a12 −a21 λ − a22 = λ2 − (a11 + a22 )λ + (a11 a22 − a12 a21 ). Consequently, the eigenvalues of A are given by λ= a11 + a22 ± 2 (a11 + a22 )2 − (a11 a22 − a12 a21 ). 4 Therefore, the eigenvalues of A are purely imaginary if and only a11 + a22 = 0 (a11 + a22 )2 − (a11 a22 − a12 a21 ) < 0. 4 and This is equivalent to saying that a11 + a22 = 0 and a11 a22 − a12 a21 > 0. (b) We now assume that (x(t), y (t)) is a solution of the ODE dx = a11 x + a12 y dt dy = a21 x + a22 y. dt In the next step, we express y (t) as a function of x(t). The resulting function y (x) sat...
View Full Document

Ask a homework question - tutors are online