Question

Consider the differential equation y' (t) = (y-2)(1 + y). a) Find the solutions that are constant, for all t20 (the equilibrium solutions). b) In what regions are solutions increasing? Decreasing? c) Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d) Sketch the direction field and verify that it is consistent with parts a through c. a) The solutions are constant for (Type an equation. Use a comma to separate answers as needed.) b) Where are the solutions increasing? A. y>0 B. y> 2 and y< -1 OC. y<-1 D. \y<3 Where are the solutions decreasing? O A. y> -1 B. -1<y<2 OC. y] >3 D. y<0

Answer 1

quien : dy = (4-2)(1+y) dt for constant solution Y=0 T 0 (4-2)(1+y) = 0 Y=2 y=-1 6) incre reasing solution and decreasing solution. g'(t) > (4-2)(1+y) >o 4-2 > y » 2 and i+y>o and boboceed Y3-1 increasing: Decreasing: y>2 and Y<-1 -Iky<2 option (B) option (B)

(0) initial conditions lead to solutions: uncreasing and decrasing time y(o) E A increasing A>2 and AS-1 option CD decreasing -IKA<2 option (A)

D) Graph . Option (B) is correct

-5 (0,2) -5 0 10 (0, -1) -5 -10