Answer to Question #203637 in Calculus for Anuj

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

Given that, 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.

(a) YES, when upward and downward slopes are at 45° then it must be the case of multiple equilibria.

(b) If the initial value y0 lies to the left of L, independent of time path will be obtained.

(c) If the initial value lies between Land R then it may be always Stability.

(d) If the initial value lies to the right of R then it may be sometimes Instability.

(e) The dynamic stability of equilibrium at L and at R are stables