Question

Consider the following differential equation date = y2(y2 – 4). (a) Find all critical values. (b) Draw the phase diagram to classify each as stable, semi-stable or unstable.

Answer 1

differential equation is given dy = y(y = 4) da (a) To find critical value, make dy da ie gely=4)=0 y=0,0 y2-4=0 y²4 = 12 y = 54 y=0,2,-2. o critical values are (6) ek at 2 dy = positive unstable - Negative semi stable. dy = Negative dy at dt stable -2 dy = at = positive J Pee -2

this point is unstable. at y=-2, dy e values are apooching aie arrows are indicated in above figure. so -a is stable *不 2 0 at y=0, de fete values from oto a approching o but 0-2 leaving so this point is semi stable t approaching 2 } leaving at y= ta dy of values are leaving 30 ket 2