MATH 61CM Study Guide - Midterm Guide: Tibet, Uniform Boundedness, Compact Space
Document Summary
Both the initial condition and ode are readily checked. (ii). Suppose y1(t), y2(t) are both solutions to the ode y (t) = a(t) y(t) with the same initial condition y(0) = y0. We claim the set {t : y1(t) = y2(t)} is the entire real line. To this end, because it is nonempty, it suf ces to show that it is both closed and open: to show the closed property, observe y1(t), y2(t) are both continuous. But y1, y2 satisfy the same ode everywhere with the same value at t, so local uniqueness provides us that y2 = y2 on some open neighborhood of this "new initial condition" t, which proves the openness. Suppose that y(t ) 6 0 for some t r. because y(t) is continuous, for some t0 between 0 and t we know y(t0) = 0; this is consequence of the intermediate value theorem.
