Problems 35–37 show that all nontrivial solution curves of the system in Example 11 are ellipses rotated by the same angle as the trajectory in Fig. 7.4.11.

It can be further shown that Eq. (65) represents in general a conic section rotated by the angle θ given by

Show that this formula applied to Eq. (64) leads to the angle θ = arctan 2/4 found in Example 11, and thus conclude that all elliptical solution curves of the system in Example 11 are rotated by the same angle θ.

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Example 11 Solve the initial value problem 6 -17 X' x = [$ ²] x x = [] x(0) 8 (46)