Lesson test: Systems and the phase plane

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Systems and the phase plane

1.[2p]
What is gained by rewriting $y'' + 3 y' + 2 y = \cos t$ as a pair of first order equations?

The state becomes an explicit pair of numbers, which is the form solvers and phase portraits need

Correct

The equation becomes linear

The answer is: The state becomes an explicit pair of numbers, which is the form solvers and phase portraits need

The initial conditions are no longer needed

The answer is: The state becomes an explicit pair of numbers, which is the form solvers and phase portraits need

The order of the equation is reduced without changing the number of constants

The answer is: The state becomes an explicit pair of numbers, which is the form solvers and phase portraits need

2.[2p]
For the system $x' = 2 x + y$ , $y' = - 3 x + 4 y$ , what is the determinant $Δ = a d - b c$ ?

CorrectNot quite: 11

3.[3p]
For the system $x' = x + 2 y$ , $y' = 3 x + 2 y$ , what is the larger eigenvalue?

CorrectNot quite: 4

4.[2p]
Why can two trajectories in the phase plane never cross?

A crossing point would have two different futures, which uniqueness forbids

Correct

The trajectories are always closed curves

The answer is: A crossing point would have two different futures, which uniqueness forbids

The velocity vector is never zero

The answer is: A crossing point would have two different futures, which uniqueness forbids

Crossing would violate conservation of energy

The answer is: A crossing point would have two different futures, which uniqueness forbids

5.[3p]
Match each condition on the trace and determinant to the type of equilibrium.

$Δ < 0$
$Δ > 0$ and $τ^{2} > 4 Δ$
$Δ > 0$ and $τ^{2} < 4 Δ$
$τ = 0$ and $Δ > 0$

node
centre
saddle
spiral

Show the answer

$Δ < 0$ : saddle $Δ > 0$ and $τ^{2} > 4 Δ$ : node $Δ > 0$ and $τ^{2} < 4 Δ$ : spiral $τ = 0$ and $Δ > 0$ : centre

6.[2p]
A centre survives a small change in the coefficients of the system.

True

The answer is: False

False

Correct

7.[3p]
The oscillator system $x' = v$ , $v' = - ω_{0}^{2} x - 2 ζ ω_{0} v$ has $ω_{0} = 5$ rad s⁻¹ and $ζ = 0.2$ . What is the trace?

CorrectNot quite: -2

8.[3p]
Which statements about linear systems in the plane are correct?

Select all that apply

Stability requires $τ < 0$ and $Δ > 0$

Correct

An eigenvector direction is one along which the motion is purely inward or outward

Correct

After a while a stable node is dominated by the eigenvalue nearest zero

Correct

A saddle can be stable if both eigenvalues are large

The answer is: Stability requires $\tau < 0$ and $\Delta > 0$, An eigenvector direction is one along which the motion is purely inward or outward, After a while a stable node is dominated by the eigenvalue nearest zero

9.[3p]
Two coupled tanks have eigenvalues $- 0.029$ and $- 0.171$ per minute. What happens after about twenty minutes?

Only the slow mode is left, so the tanks drain together in a fixed proportion

Correct

The tanks reach equilibrium with equal salt masses

The answer is: Only the slow mode is left, so the tanks drain together in a fixed proportion

The fast mode dominates, since its eigenvalue is larger in magnitude

The answer is: Only the slow mode is left, so the tanks drain together in a fixed proportion

The system becomes unstable

The answer is: Only the slow mode is left, so the tanks drain together in a fixed proportion

1.[2p] What is gained by rewriting y''+3y'+2y=cost as a pair of first order equations?

2.[2p] For the system x'=2x+y, y'=-3x+4y, what is the determinant Δ=ad-bc?

3.[3p] For the system x'=x+2y, y'=3x+2y, what is the larger eigenvalue?

4.[2p] Why can two trajectories in the phase plane never cross?

5.[3p] Match each condition on the trace and determinant to the type of equilibrium.

6.[2p] A centre survives a small change in the coefficients of the system.

7.[3p] The oscillator system x'=v, v'=-ω02x-2ζω0v has ω0=5 rad s⁻¹ and ζ=0.2. What is the trace?

8.[3p] Which statements about linear systems in the plane are correct?

9.[3p] Two coupled tanks have eigenvalues -0.029 and -0.171 per minute. What happens after about twenty minutes?

1.[2p]
What is gained by rewriting $y'' + 3 y' + 2 y = \cos t$ as a pair of first order equations?

2.[2p]
For the system $x' = 2 x + y$ , $y' = - 3 x + 4 y$ , what is the determinant $Δ = a d - b c$ ?

3.[3p]
For the system $x' = x + 2 y$ , $y' = 3 x + 2 y$ , what is the larger eigenvalue?

4.[2p]
Why can two trajectories in the phase plane never cross?

5.[3p]
Match each condition on the trace and determinant to the type of equilibrium.

6.[2p]
A centre survives a small change in the coefficients of the system.

7.[3p]
The oscillator system $x' = v$ , $v' = - ω_{0}^{2} x - 2 ζ ω_{0} v$ has $ω_{0} = 5$ rad s⁻¹ and $ζ = 0.2$ . What is the trace?

8.[3p]
Which statements about linear systems in the plane are correct?

9.[3p]
Two coupled tanks have eigenvalues $- 0.029$ and $- 0.171$ per minute. What happens after about twenty minutes?