We can give a simpler description of the general solution

Chapter , Problem 29

(choose chapter or problem)

We can give a simpler description of the general solution x.t / D c1 1 6 e2t C c2 1 1 e5t (9) of the system x0 D 4 1 6 1 x in Example 1 by introducing the oblique uv-coordinate system indicated in Fig. 5.3.17, in which the u-and vaxes are determined by the eigenvectors v1 D 1 6 and v2 D 1 1 , respectively. v1 v2 u v u v FIGURE 5.3.17. The oblique uv-coordinate system determined by the eigenvectors v1 and v2. The uv-coordinate functions u.t / and v.t / of the moving point x.t / are simply its distances from the origin measured in the directions parallel to v1 and v2. It follows from (9) that a trajectory of the system is described by u.t / D u0e2t ; v.t / D v0e5t (63) where u0 D u.0/ and v0 D v.0/. (a) Show that if v0 D 0, then this trajectory lies on the u-axis, whereas if u0 D 0, then it lies on the v-axis. (b) Show that if u0 and v0 are both nonzero, then a Cartesian equation of the parametric curve in Eq. (63) is given by v D C u5=2. 30.