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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- Fall '11
- Economics, Orthogonal matrix, Tangent bundle, first-order differential equations