I a it can be shown that the partial sums on the left

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Unformatted text preview: hence the functions are linearly independent at every point. 28 . Let and . It follows that and . In terms of the new variables, we obtain the system of two first order ODEs . The associated eigenvalue problem is . The characteristic equation is , with roots . Since the eigenvalues are purely imaginary, the origin is a center. Hence the phase curves are ellipses, with a clockwise flow. For computational purposes, let and . ________________________________________________________________________ page 401 —————————————————————————— CHAPTER 7. —— . The general solution of the second order equation is . The general solution of the system of ODEs is given by ________________________________________________________________________ page 402 —————————————————————————— CHAPTER 7. —— x It is evident that the natural frequency of the system is equal to . ______________________________________...
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