In this problem we indicate how to show that the trajectories are ellipses when

Chapter 9, Problem 19

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In this problem we indicate how to show that the trajectories are ellipses when theeigenvalues are pure imaginary. Consider the systemxy=a11 a12a21 a22 xy. (i)(a) Show that the eigenvalues of the coefficient matrix are pure imaginary if and only ifa11 + a22 = 0, a11a22 a12a21 > 0. (ii)(b) The trajectories of the system (i) can be found by converting Eqs. (i) into the singleequationdydx = dy/dtdx/dt = a21x + a22ya11x + a12y. (iii)Use the first of Eqs. (ii) to show that Eq. (iii) is exact.(c) By integrating Eq. (iii), show thata21x2 + 2a22xy a12y2 = k, (iv)where k is a constant. Use Eqs. (ii) to conclude that the graph of Eq. (iv) is always anellipse.Hint: What is the discriminant of the quadratic form in Eq. (iv)?