Therefore the general solution is imposing the initial

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Unformatted text preview: _______________ page 391 —————————————————————————— CHAPTER 7. —— 14 . The roots of the characteristic equation, . Note that the roots are complex when , the equilibrium point when , are . For the case when is a stable spiral. On the other hand, , the equilibrium point is an unstable spiral. For the case , the roots are purely imaginary, so the equilibrium point is a center. When , the roots are real and distinct. The equilibrium point becomes a node, with its stability dependent on the sign of Finally, the case marks the transition from spirals to nodes. . ________________________________________________________________________ page 392 —————————————————————————— CHAPTER 7. —— 17. The characteristic equation of the coefficient matrix is , with roots given formally as . The roots are real provided that First note that the sum of the roots is and the product of the roots is . For negative va...
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