- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Logistic growth and phase lines practice question 1
How many steady states does this differential equation have?
A steady state is a value of at which . Factor the right-hand side completely before solving.
Take out the common factor rather than dividing by it, then factor what is left. Count each distinct value of once.
A whole number.
Theory UBC Math 100 §13.1Key results
Worked solution
Strategy
The steady states are the zeros of the right-hand side. Factor it fully and count the distinct values of y that make it zero.
Working
Factor out y² rather than dividing by it, so the root y = 0 is kept.
A product is zero exactly when one of its factors is.
The squared factor still gives only one value of y, so there are three.
Answer
The answer is three steady states, at y = −2, 0 and 2.
Check
As a quadratic in y², y⁴ − 4y² is zero only for y² = 0 or y² = 4, and those give the three values 0, 2 and −2.
Where this goes wrong
Dividing both sides by y² to simplify, which throws away the steady state y = 0. Factor instead, and count each distinct zero of the right-hand side once.
Leans on
Factoring a polynomial, from Functions, graphs and inverses
Saved in this browser only.