Question

Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k > 0

Answer 1

dy a) The critical Points for = y(k - y), t> 0, k > 0 dt are given by y(k – y) =0 or, y=0 and y=k b) у 110 y=k Stable dy dt = y(k - y) t > y=0 Un stable -15 -10 0 10 1 +10 1 T 1 C dy = y(k- y) or, dt = k-2y or, dy -2+0 dt -2 dt? d'y For y<0, di? k dy 20=k-2y20=0sy< dit? d'y <O>>y<k 2 d’y For y>K, k dt2 k Hence The curve is concave up for y>K and Osy< 2 k The curve is concave down for y<0 and <y<k 2 d) у 110 - - 1 - - 1 D(3.58, 7.47) 1 - B(1.38, 3:04) t -10 10 - +10