Solution Found!
Assuming that the trajectory corresponding to a solution x = (t), y = (t), < t < , of an
Chapter 9, Problem 24(choose chapter or problem)
Assuming that the trajectory corresponding to a solution x = (t), y = (t), < t < , of an autonomous system is closed, show that the solution is periodic. Hint: Since the trajectory is closed, there exists at least one point ( x0, y0) such that (t0) = x0, (t0) = y0 and a number T > 0 such that (t0 + T ) = x0, (t0 + T ) = y0. Show that x = (t) = (t + T ) and y = (t) = (t + T ) is a solution, and then use the existence and uniqueness theorem to show that (t) = (t) and (t) = (t) for all t.
Questions & Answers
QUESTION:
Assuming that the trajectory corresponding to a solution x = (t), y = (t), < t < , of an autonomous system is closed, show that the solution is periodic. Hint: Since the trajectory is closed, there exists at least one point ( x0, y0) such that (t0) = x0, (t0) = y0 and a number T > 0 such that (t0 + T ) = x0, (t0 + T ) = y0. Show that x = (t) = (t + T ) and y = (t) = (t + T ) is a solution, and then use the existence and uniqueness theorem to show that (t) = (t) and (t) = (t) for all t.
ANSWER:Step 1 of 3
Given the system and
And a number such that
,