Solve problem 1.30 from Advanced differential equations . Question is based on Qualitative analysis of first order equations.

Main topic is ordinary differential equations

he notion of a super solution by requiring only r+ (t) 2 f (t, x+(t)). Then a+(to) 2 x(to) implies a+ (t) 2 x(t) for all t 2 to and if strict inequality becomes true at some time it remains true for all later times. Problem 1.27. Let x be a solution of (1.61) which satisfies limt-x x(t) = 1. Show that limt x x(t) = 0 and f(x1) = 0. (Hint: If you prove lime-xi(t) = 0 without using (1.61) your proof is wrong! Can you give a counter example?) Problem 1.28. Prove Lemma 1.1. (Hint: This can be done either by using the analysis from Section 1.3 or by using the previous problem.) Problem 1.29. Generalize the concept of sub and super solutions to the interval (T, to), where T