(Solution) - For a 2 x 2 linear DE system we can -(2025 Original AI-Free Solution)

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For a 2 x 2 linear DE system

we can make a quick handsketch of the elliptical trajectories as follows.
(a) All the elliptical trajectories are concentric and similar, so we can get all the key information from just one. We have from equation (7) that
By choosing the particular solution where c1 =I, c2= 0, we reduce the solution to the single equation (6).
(b) Show that for t = 0,
Show also that for ?t = ?.
Then show that for ?t = ? / 2