Chapter 3.4, Problem 23P

Applications.

In this problem, we indicate how to show that the trajectories are ellipses when the eigenvalues are purely imaginary.

Consider the system

${\left(\begin{array}{l}x\\ y\end{array}\right)}^{\text{'}}=\left(\begin{array}{cc}{a}_{ll}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)$. (i)

(a) Show that the eigenvalues of the coefficient matrix are purely imaginary if and only if

$a_{11}+a_{22}=0,a_{11}{a}_{22}-a_{12}{a}_{21}>0$. (ii)

(b) The trajectories of the system (i) can be found by convertingEqs. (i) into the single equation

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{a_{21}x+a_{22}y}{a_{11}x+a_{12}y}$. (iii)

Use the first of Eqs. (ii) to show that Eq. (iii) is exact.

(c) By integrating Eq. (iii), show that

$a_{21}{x}^2+2a_{22}xy-a_{12}{y}^2=k$, (iv)

Where $k$ is a constant. Use Eqs. (ii) to conclude that the graph of Eq. (iv) is always an ellipse.

Hint: What is the discriminant of the quadratic form in Eq. (iv) ?