Lesson test: Autonomous equations and stability

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Autonomous equations and stability

1.[2p]
At an equilibrium $y^{}$ of $y' = f (y)$ , what does $f' (y^{}) < 0$ tell you?

The equilibrium is stable, since a small displacement decays like $e^{f' (y^{*}) t}$

Correct

The equilibrium is unstable, since $f$ is decreasing there

The answer is: The equilibrium is stable, since a small displacement decays like $e^{f'(y^*)t}$

The solution reaches the equilibrium in finite time

The answer is: The equilibrium is stable, since a small displacement decays like $e^{f'(y^*)t}$

Nothing, unless $f'' (y^{*})$ is also known

The answer is: The equilibrium is stable, since a small displacement decays like $e^{f'(y^*)t}$

2.[1p]
An autonomous first order equation can have a solution that oscillates up and down.

True

The answer is: False

False

Correct

3.[2p]
For $y' = 3 y - y^{2}$ , what is the value of the stable equilibrium?

CorrectNot quite: 3

4.[3p]
A falling body obeys $v' = g - (b / m) v^{2}$ with $m / b = 500$ m and $g = 9.81$ m s⁻². What is the terminal speed in metres per second?

CorrectNot quite: 70.0

5.[3p]
A logistic population has $r = 0.5$ per year and $K = 800$ tonnes. What is the largest constant harvest, in tonnes per year, that it can sustain?

CorrectNot quite: 100

6.[2p]
A logistic population has $r = 0.08$ per year. What is the time constant, in years, of its approach to the carrying capacity?

CorrectNot quite: 12.5

7.[3p]
Which statements about the phase line are correct?

Select all that apply

Equilibria are the roots of $f$ , and no solution can cross one

Correct

Between consecutive equilibria a solution is monotone

Correct

An equilibrium with arrows pointing in from both sides is stable

Correct

The linearisation test settles every case, including $f' (y^{*}) = 0$

The answer is: Equilibria are the roots of $f$, and no solution can cross one, Between consecutive equilibria a solution is monotone, An equilibrium with arrows pointing in from both sides is stable

8.[3p]
Why can a harvested fishery collapse with little warning as the catch is raised?

The stable and unstable equilibria collide and vanish at a critical catch, leaving no equilibrium at all

Correct

The carrying capacity falls to zero as the catch rises

The answer is: The stable and unstable equilibria collide and vanish at a critical catch, leaving no equilibrium at all

The growth rate $r$ becomes negative

The answer is: The stable and unstable equilibria collide and vanish at a critical catch, leaving no equilibrium at all

The population becomes non-autonomous

The answer is: The stable and unstable equilibria collide and vanish at a critical catch, leaving no equilibrium at all

9.[3p]
Match each equation to the character of its equilibrium at $y = 0$ .

$y' = - 2 y$
$y' = 2 y$
$y' = y^{2}$
$y' = - y^{3}$

unstable
stable
stable
semi-stable

Show the answer

$y' = - 2 y$ : stable $y' = 2 y$ : unstable $y' = y^{2}$ : semi-stable $y' = - y^{3}$ : stable

1.[2p] At an equilibrium y* of y'=f(y), what does f'(y*)<0 tell you?

2.[1p] An autonomous first order equation can have a solution that oscillates up and down.

3.[2p] For y'=3y-y2, what is the value of the stable equilibrium?

4.[3p] A falling body obeys v'=g-(b/m)v2 with m/b=500 m and g=9.81 m s⁻². What is the terminal speed in metres per second?

5.[3p] A logistic population has r=0.5 per year and K=800 tonnes. What is the largest constant harvest, in tonnes per year, that it can sustain?

6.[2p] A logistic population has r=0.08 per year. What is the time constant, in years, of its approach to the carrying capacity?

7.[3p] Which statements about the phase line are correct?

8.[3p] Why can a harvested fishery collapse with little warning as the catch is raised?

9.[3p] Match each equation to the character of its equilibrium at y=0.

1.[2p]
At an equilibrium $y^{}$ of $y' = f (y)$ , what does $f' (y^{}) < 0$ tell you?

2.[1p]
An autonomous first order equation can have a solution that oscillates up and down.

3.[2p]
For $y' = 3 y - y^{2}$ , what is the value of the stable equilibrium?

4.[3p]
A falling body obeys $v' = g - (b / m) v^{2}$ with $m / b = 500$ m and $g = 9.81$ m s⁻². What is the terminal speed in metres per second?

5.[3p]
A logistic population has $r = 0.5$ per year and $K = 800$ tonnes. What is the largest constant harvest, in tonnes per year, that it can sustain?

6.[2p]
A logistic population has $r = 0.08$ per year. What is the time constant, in years, of its approach to the carrying capacity?

7.[3p]
Which statements about the phase line are correct?

8.[3p]
Why can a harvested fishery collapse with little warning as the catch is raised?

9.[3p]
Match each equation to the character of its equilibrium at $y = 0$ .