Lesson test: Constant coefficients

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Constant coefficients

1.[2p]
Why does substituting $y = e^{r t}$ solve a constant coefficient equation?

Each derivative returns the same exponential times a power of $r$ , so the equation collapses to a polynomial in $r$

Correct

Because every solution of every linear equation is an exponential

The answer is: Each derivative returns the same exponential times a power of $r$, so the equation collapses to a polynomial in $r$

Because $e^{r t}$ satisfies the initial conditions automatically

The answer is: Each derivative returns the same exponential times a power of $r$, so the equation collapses to a polynomial in $r$

Because the Wronskian of exponentials is always one

The answer is: Each derivative returns the same exponential times a power of $r$, so the equation collapses to a polynomial in $r$

2.[3p]
For $y'' + 2 y' + 10 y = 0$ , the roots are $α \pm i β$ . What is $β$ ?

CorrectNot quite: 3

3.[2p]
For $y'' + 8 y' + 16 y = 0$ , what is the repeated root $r$ ?

CorrectNot quite: -4

4.[1p]
A repeated characteristic root leaves the equation with only one independent solution.

True

The answer is: False

False

Correct

5.[3p]
Solve $y'' + 3 y' + 2 y = 0$ with $y (0) = 1$ and $y' (0) = 0$ . What is $y (1)$ ?

CorrectNot quite: 0.600

6.[3p]
The roots of a characteristic equation are $- 0.5 \pm 4 i$ . What does the solution do?

It oscillates at angular frequency 4 inside an envelope decaying with time constant 2

Correct

It decays without oscillating, with time constant 2

The answer is: It oscillates at angular frequency 4 inside an envelope decaying with time constant 2

It oscillates with growing amplitude

The answer is: It oscillates at angular frequency 4 inside an envelope decaying with time constant 2

It approaches a nonzero constant

The answer is: It oscillates at angular frequency 4 inside an envelope decaying with time constant 2

7.[3p]
Match each root pattern to the behaviour of the general solution.

Two distinct negative real roots
Complex pair with negative real part
Purely imaginary pair
Any root with positive real part

decay without oscillation
undamped oscillation
unbounded growth
decaying oscillation

Show the answer

Two distinct negative real roots: decay without oscillation Complex pair with negative real part: decaying oscillation Purely imaginary pair: undamped oscillation Any root with positive real part: unbounded growth

8.[3p]
Which statements are correct for constant coefficient equations?

Select all that apply

A root repeated $k$ times contributes solutions multiplied by $1, t, \dots, t^{k - 1}$

Correct

An $n$ -th order equation has a solution space of dimension $n$

Correct

Complex roots give real solutions once Euler's formula is applied

Correct

Characteristic polynomials of any degree can be solved by a formula in the coefficients

The answer is: A root repeated $k$ times contributes solutions multiplied by $1, t, \ldots, t^{k-1}$, An $n$-th order equation has a solution space of dimension $n$, Complex roots give real solutions once Euler's formula is applied

9.[3p]
The Cauchy-Euler equation $t^{2} y'' + 2 t y' - 6 y = 0$ has solutions $t^{r}$ . What is the positive value of $r$ ?

CorrectNot quite: 2

1.[2p] Why does substituting y=ert solve a constant coefficient equation?

2.[3p] For y''+2y'+10y=0, the roots are α±iβ. What is β?

3.[2p] For y''+8y'+16y=0, what is the repeated root r?

4.[1p] A repeated characteristic root leaves the equation with only one independent solution.

5.[3p] Solve y''+3y'+2y=0 with y(0)=1 and y'(0)=0. What is y(1)?

6.[3p] The roots of a characteristic equation are -0.5±4i. What does the solution do?

7.[3p] Match each root pattern to the behaviour of the general solution.

8.[3p] Which statements are correct for constant coefficient equations?

9.[3p] The Cauchy-Euler equation t2y''+2ty'-6y=0 has solutions tr. What is the positive value of r?

1.[2p]
Why does substituting $y = e^{r t}$ solve a constant coefficient equation?

2.[3p]
For $y'' + 2 y' + 10 y = 0$ , the roots are $α \pm i β$ . What is $β$ ?

3.[2p]
For $y'' + 8 y' + 16 y = 0$ , what is the repeated root $r$ ?

4.[1p]
A repeated characteristic root leaves the equation with only one independent solution.

5.[3p]
Solve $y'' + 3 y' + 2 y = 0$ with $y (0) = 1$ and $y' (0) = 0$ . What is $y (1)$ ?

6.[3p]
The roots of a characteristic equation are $- 0.5 \pm 4 i$ . What does the solution do?

7.[3p]
Match each root pattern to the behaviour of the general solution.

8.[3p]
Which statements are correct for constant coefficient equations?

9.[3p]
The Cauchy-Euler equation $t^{2} y'' + 2 t y' - 6 y = 0$ has solutions $t^{r}$ . What is the positive value of $r$ ?