Consider the equation
𝑑
𝑦
𝑑
𝑡
=
𝑦
2
(
25
−
𝑦
2
)
. Which of the following are true?

(1)
𝑦
=
0
is an (asymptotically) unstable equilibrium point.

(2)
𝑦
=
0
is a (asymptotically) stable equilibrium point.

(3)
𝑦
=
0
is a (asymptotically) semistable equilibrium point.

(4)
𝑦
=
−
5
is an (asymptotically) unstable equilibrium point.

(5)
𝑦
=
5
is a (asymptotically) stable equilibrium point.

Question

Consider the equation 
𝑑
𝑦
𝑑
𝑡
=
𝑦
2
(
25
−
𝑦
2
)
.   Which of the following are true?

(1) 
𝑦
=
0
  is an (asymptotically) unstable equilibrium point.

(2)  
𝑦
=
0
  is a (asymptotically) stable equilibrium point.

(3) 
𝑦
=
0
 is a (asymptotically) semistable equilibrium point.

(4) 
𝑦
=
−
5
 is an (asymptotically) unstable equilibrium point.

(5) 
𝑦
=
5
  is a (asymptotically) stable equilibrium point.

BlackTom AI · Accepted Answer

We begin by identifying the equilibria of dy/dt = y^2(25 − y^2). Setting the right-hand side to zero gives y = 0 and y = ±5 as equilibrium points. Option (1): y = 0 as an asymptotically unstable equilibrium. To judge stability, consider small perturbations around 0. For small y ≠ 0, dy/dt ≈ 25 y^2 > 0, so any nonzero y near 0 tends to increase. If y > 0, it moves away from 0 toward larger positive values; if y < 0, dy/dt is still positive and y increases toward 0, but does not cross and continue to stay near or move away depending on the trajectory. The net behavior indicates that zero is not asymptotically unstable in the typical sense, and in fact the analysis shows a semi-stable behavior rather than simple asymptotic instability. Therefore this statement is not correct. Option (2): y = 0 as a (asymptotically) stable equilibrium. Since small positive y grows (dy/dt > 0) and tends to move away from 0, whereas small negative y increases toward 0 but does not settle there for all directions, the equilibrium is not asymptotically stable from all perturbations. Hence this statement is false. Option (3): y = 0 is a semistable equilibrium point. As discussed, for tiny positive perturbations, the solution moves away from 0 toward the positive side (toward +5), whereas for tiny negative perturbations, the solution increases toward 0 but from the negative side it tends to approach 0 in the short term and then moves away in the long term depending on the trajectory. This asymmetric behavior characterizes semistability. Thus this statement is true. Option (4): y = −5 is an unstable equilibrium point. Evaluate stability via f'(y) at the equilibrium: f(y) = y^2(25 − y^2); f'(y) = 2y(25 − y^2) − 2y^3 = 50y − 4y^3. At y = −5, f'(−5) = 50(−5) − 4(−125) = −250 + 500 = 250 > 0, indicating instability. Therefore this statement is true. Option (5): y = 5 is a stable equilibrium point. Using the same derivative, f'(5) = 50(5) − 4(125) = 250 − 500 = −250 < 0, which corresponds to asymptotic stability. Hence this statement is true. In summary, the true statements are those asserting semistability at 0 and instability at −5 and stability at 5, i.e., options (3), (4), and (5) are correct. The remaining statements (1) and (2) misclassify the nature of the equilibrium at 0.

Similar Questions

How did Egyptian culture and governance change during the Ptolemaic period compared to the pharaonic period?

Name the Supreme Court decision that removed financial constraints on PACs.

Political scientists Joshua Kalla and David Broockman find that senior policy makers made themselves available to _____ between three and four times more often than they did to constituents.

Name the political scientist who argues that parties are designed to serve the needs of ambitious politicians (namely candidates and officeholders).

Name the political scientist who said this famous quote about the problems of pluralism: “The flaw in the pluralist heaven is that the heavenly choir sings with a strong upper-class accent.”

What is the name of the third party candidate in 2000 who was not allowed to participate in the presidential debates? Hint: We watched a video clip with him in class.

E.E. Schattshneider said, "Democracy is ________ save in terms of the parties."

Who was the most recently appointed justice to the Supreme Court?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler