Lesson test: Linear equations and the integrating factor

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Linear equations and the integrating factor

1.[2p]
What property is the integrating factor $μ$ chosen to give?

It makes the left side exactly the derivative of the product $μ y$

Correct

It cancels the forcing term $g (t)$

The answer is: It makes the left side exactly the derivative of the product $μ y$

It makes the equation separable

The answer is: It makes the left side exactly the derivative of the product $μ y$

It removes the arbitrary constant

The answer is: It makes the left side exactly the derivative of the product $μ y$

2.[2p]
What is the integrating factor for $y' + \frac{3}{t} y = t$ ?

$μ = t^{3}$

Correct

$μ = e^{3 t}$

The answer is: $μ = t^{3}$

$μ = 3 \ln t$

The answer is: $μ = t^{3}$

$μ = e^{t^{3}}$

The answer is: $μ = t^{3}$

3.[3p]
Solve $y' + 2 y = 4$ with $y (0) = 1$ . What is $y (1)$ ?

CorrectNot quite: 1.865

4.[1p]
For $y' + y / τ = g$ with $g$ constant, the steady state depends on the initial condition.

True

The answer is: False

False

Correct

5.[3p]
A resistor and capacitor with time constant 2.0 s charge from a 12 V supply, starting at 0 V. After how many seconds does the capacitor reach 8 V?

CorrectNot quite: 2.20

6.[3p]
A 200 litre tank of pure water receives brine at 0.1 kg per litre flowing at 5 litres per minute, with the mixture leaving at the same rate. How many kilograms of salt are in the tank after 40 minutes?

CorrectNot quite: 12.6

7.[3p]
Which statements about the solution of a first order linear equation are correct?

Select all that apply

It splits into a transient carrying the initial condition and a response to the forcing

Correct

The transient decays with time constant $1 / p$ when $p$ is a positive constant

Correct

The steady state is set by the forcing and the coefficient

Correct

The transient determines the long-term behaviour

The answer is: It splits into a transient carrying the initial condition and a response to the forcing, The transient decays with time constant $1 / p$ when $p$ is a positive constant, The steady state is set by the forcing and the coefficient

8.[2p]
A low-pass filter driven at $ω τ = 1$ passes what fraction of the input amplitude?

CorrectNot quite: 0.707

9.[3p]
Match each item to what it is.

$e^{\int p d t}$
$V / q$ for a stirred tank
$v = y^{1 - n}$
$R C$ for a series circuit

the integrating factor
the time constant
the time constant
the Bernoulli substitution

Show the answer

$e^{\int p d t}$ : the integrating factor $V / q$ for a stirred tank: the time constant $v = y^{1 - n}$ : the Bernoulli substitution $R C$ for a series circuit: the time constant

1.[2p] What property is the integrating factor μ chosen to give?

2.[2p] What is the integrating factor for y′+3ty=t?

3.[3p] Solve y′+2y=4 with y(0)=1. What is y(1)?

4.[1p] For y′+y/τ=g with g constant, the steady state depends on the initial condition.

5.[3p] A resistor and capacitor with time constant 2.0 s charge from a 12 V supply, starting at 0 V. After how many seconds does the capacitor reach 8 V?

6.[3p] A 200 litre tank of pure water receives brine at 0.1 kg per litre flowing at 5 litres per minute, with the mixture leaving at the same rate. How many kilograms of salt are in the tank after 40 minutes?

7.[3p] Which statements about the solution of a first order linear equation are correct?

8.[2p] A low-pass filter driven at ωτ=1 passes what fraction of the input amplitude?

9.[3p] Match each item to what it is.

1.[2p]
What property is the integrating factor $μ$ chosen to give?

2.[2p]
What is the integrating factor for $y' + \frac{3}{t} y = t$ ?

3.[3p]
Solve $y' + 2 y = 4$ with $y (0) = 1$ . What is $y (1)$ ?

4.[1p]
For $y' + y / τ = g$ with $g$ constant, the steady state depends on the initial condition.

5.[3p]
A resistor and capacitor with time constant 2.0 s charge from a 12 V supply, starting at 0 V. After how many seconds does the capacitor reach 8 V?

6.[3p]
A 200 litre tank of pure water receives brine at 0.1 kg per litre flowing at 5 litres per minute, with the mixture leaving at the same rate. How many kilograms of salt are in the tank after 40 minutes?

7.[3p]
Which statements about the solution of a first order linear equation are correct?

8.[2p]
A low-pass filter driven at $ω τ = 1$ passes what fraction of the input amplitude?

9.[3p]
Match each item to what it is.