Answer to Question #203430 in Calculus for Aryan

Question

As a phase line, use a U-shaped curve. Let its downward-sloping segment intersect the 45 degree line at point L, and let its upward-sloping segment intersect the 45° line at point R. Answer the following five questions:

(a) Is this a case of multiple equilibria?

(b) If the initial value y₀ lies to the left of L, what kind of time path will be obtained?

(c) What if the initial value lies between Land R?

(d) What if the initial value lies to the right of R?

(e) What can you conclude about the dynamic stability of equilibrium at L and at R, respectively? 

Answer 1

Answer:-

U-shaped curve

downward sloping segment intersect 45^o line at point L,

its upward sloping segment intersect 45^o line at point R.

(a) Yes when upward and downward sloping at 45^o than it must be the

case of multiple equilibria.(b) the independent time parts will find

having initial value y_o

(c) if initial value lies between L and R than it may be always stability condition.

(d) if values are right of R then it may be sometimes unstable

(e) we can conclude that the stability is dependent on L and R , according to initial value we obtain different result as stated in above points