Question

Consider the differential equation y' (t) = (y-4)(1 + y). a) Find the solutions that are constant, for all t2 0 (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? O A. y>4 and y< - 1 OB. ly<5 O C. y>0 OD. y<-1 Where are the solutions decreasing? O A. y> -1 OB. y< 0 OC. jy|25 OD. -1<y<4 c) Which initial conditions lead to solutions that are increasing in time? Click to select your answer(s).

Answer 1

Solution - Given differential equation YiCt) = (y-u) Ly+1) For equilibrium solution- (+)= 0 ) 4-45 0 0 +1=0 > y = 4 and y=-11 Put 4:4 y = -1 © - The solution are cmireasing whese y 34 and y2-1 (from graph) » The sofution are decreasing where -12424 ② - A >4 and AL-1 Coption A) -KAL 4 (obtion A) obtim