I am stuck with an exercise.
Let $ f = f(t, y) $ be a continuous function such that $ \partial f / \partial y $ is continuous in all points $ (t, y) $. Say what you can about existence of global solution, sign and other estimations for the ODE $ dy/dt = f(t,y) $ under the following hypothesis:
a) Initial condition $ y(0) = 1 $, and $ y_1(t) = 3 $ is a solution for all $ t $.
b) Initial condition $ y(0) = 0 $, and $ y_1(t) =-1 $ and $ y_2(t) = e - t² + 1 $ are solutions for all $ t $.
c) Initial condition $ y(0) = 1 $, and $ y_1(t) = t + 2 $ and $ y_2(t) = -t² $ are solutions for all $ t $.
I know any initial condition $ y(t_0) = y_0 $ gives an unique solution to the problem, because $ f $ and its derivative $ \partial f / \partial y $ are continuous on the entire plane. My question is, what more can I say?
I don't expect a complete solution of the exercise, only some hints about how to go on.
In a scalar ODE with the given assumptions, if $y_1(0)<y_2(0)$ then $y_1(t)<y_2(t)$ for all times where both solutions exist, simply because solutions can not cross under the uniqueness condition.
Furthermore, if a solution is bounded to both sides, then its maximal domain is at least as large as the intersection of the domains of the bounds.