We can give a simpler description of the general solution

of the system

in Example 1 by introducing the oblique uv-coordinate system indicated in Fig. 7.4.17, in which the u- and v-axes are determined by the eigenvectors v₁ = [-1 6] and v2 [1 1] , respectively.

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

where u₀ = u(0) and v₀ = v(0).

(a) Show that if v₀ = 0, then this trajectory lies on the u-axis, whereas if u₀ = 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 = Cu^-5/2.

Transcribed Image Text:

21 X(t) = ₁₁ [~ 1 ] e ²² C1 5t €₂ [1] est +62 (9)