Economics Dynamics Problems 189

Economics Dynamics Problems 189 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 173 Figure 4.21. Figure 4.22. of at the beginning of section 4.6. The direction field, along with the independent vectors is shown in figure 4.22 for this example. For the second sub-case, where again r < 0, for large t the dominant term must be c2 ert tv, and hence as t → ∞ every trajectory must approach the origin and in such a manner that it is tangent to the line through the eigenvector v. Certainly, if c2 = 0 then the solution must lie on the line through the eigenvector v, and approaches the origin along this line, as shown in figure 4.23. (Had r > 0, then every trajectory would have moved away from the origin.) The approach of the trajectories to the origin depends on the eigenvectors v and v2 . One possibility is illustrated in figure 4.23. To see what is happening, express ...
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