Question

Consider the autonomous first-order differential equation y = 10 + 3y – v2 Find the DISTINCT critical points and classify each as (1) AS for Asymptotically Stable, (2) US for Unstable or (3) SS for Semi-Stable. Enter your answer as a comma separated list of pairs consisting on a critical point and its stability type (e.g. your answer might look like (2,AS), (-3,SS), (7,US)) Critical Point and Stability: For the initial value problem y' = 10 + 3y – y, y(0) = 3 we have lim y(x) = x00

Answer 1

Sola: y'= 10 + 3y - y" - fly) Critical points occur whenever fly) = 0 10+ 3y - y = 0 = 10 + Sy– 2y - y = 0 = 5(2+y) – (2+ y) = 0 = (5-y)(2+y) = 0 7 y=-2,5 are coitical points. At y=-2 : Foy y<-2 fly) < 0 Fox -22y <5; fly) > 0 the equiliboium som y=-2 is unstable At y=5 : For - For . the equilibrium -2<y<5, f(y) >0 y 75 , fly) < 0 som y=5 is asymptotically stable . For the y) is shown initial value plot in below. problem y(o) = 3, the this corresponding direction soms field

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ------ ta 5 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII HULLIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 4111711-HUH-1HH10 HUHHH?! HAHH4 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH 1111111111111111111111111111111111111 HUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH From the above graph, lim ya) = 5 uto