MAT 22B: PROBLEM SET 1 3 Problem 6. (20 pts) Consider the following differential equation: 2/6) : 2t(1 - EH2: t e R (a) Find all solutions of the differential equation above.
(b) Does there exist a solution y1(t) such that y1(2) = 1 and y1(3) = 1 ?
(0) Plot qualitatively the graph of six different solutions of the differential equation. (d) Find the long-term behaviour of all solutions for the differential equation. Problem 7. (20 pts) Consider the following differential equation: Mt) = y(t)(1- 905)): t E 13+,
(a) Find all solutions of the differential equation above. (b) Describe the possible long—term behaviour of a solution y(t) in terms of its ini-
tial value “9(0). (0) Compare the differential equation above with the differential equation 3/6) = 31(15): t6 R“, as if the}r were two models for population growth, i.e. y(t) is the size of a
population at time t E R. Which of the two models is more accurate 7
(d) Consider the differential equation
9’05) = A -y(t)(1 - Mil): 756 RJ’A E W’- How do the solutions differ for distinct values of A E R ?