Applications.
In this problem, we indicate how to show that the trajectories are ellipses when the eigenvalues are purely imaginary.
Consider the system
(a) Show that the eigenvalues of the coefficient matrix are purely imaginary if and only if
(b) The trajectories of the system (i) can be found by convertingEqs. (i) into the single equation
Use the first of Eqs. (ii) to show that Eq. (iii) is exact.
(c) By
Where
Hint: What is the discriminant of the quadratic form in Eq. (iv) ?
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Differential Equations: An Introduction to Modern Methods and Applications
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