# Quiz 11: Second Order Linear Differential Equations

## Question 1

If $y={e}^{2t}$ is a solution
to $\frac{{d}^{2}y}{d{t}^{2}}-5\frac{dy}{dt}+ky=0$, what is
the value of $k$?

**Your answer is correct**

**Not correct. You may try again.**

## Question 2

If $y={e}^{3x}cosx$ is a solution
to $\frac{{d}^{2}y}{d{x}^{2}}-6\frac{dy}{dx}+ky=0$, what is
the value of $k$?

**Your answer is correct**

**Not correct. You may try again.**

## Question 3

Which of the following is the general solution to
$\frac{{d}^{2}y}{d{x}^{2}}+3\frac{dy}{dx}-10y=0$?

In each case, $A$ and $B$ are arbitrary constants.

In each case, $A$ and $B$ are arbitrary constants.

**Your answer is correct.**

**Not correct. Choice (b) is false.**

Note the roots of the auxiliary equation are 2 and
$-5$.

**Not correct. Choice (c) is false.**

Note the roots of the auxiliary equation are 2 and
$-5$.

**Not correct. Choice (d) is false.**

Note that the roots of the auxiliary equation are 2 and
$-5$.

## Question 4

Which of the following is the general solution to
$\frac{{d}^{2}y}{d{t}^{2}}+4\frac{dy}{dt}-4y=0$?

In each case, $A$ and $B$ are arbitrary constants.

In each case, $A$ and $B$ are arbitrary constants.

**Not correct. Choice (a) is false.**

Note that the roots of the auxiliary equation are
$-2\pm 2\sqrt{2}$.

**Not correct. Choice (b) is false.**

Note that the roots of the
auxiliary equation are $-2\pm 2\sqrt{2}$.

**Not correct. Choice (c) is false.**

Note that the roots of the
auxiliary equation are $-2\pm 2\sqrt{2}$.

**Your answer is correct.**

## Question 5

Which of the following are general solutions to
$\frac{{d}^{2}x}{d{t}^{2}}-4\frac{dx}{dt}+13x=0$?

In each case, $A$ and $B$ are arbitrary constants. More than one option may be correct.

In each case, $A$ and $B$ are arbitrary constants. More than one option may be correct.

**There is at least one mistake.**

For example, choice (a) should be false.

**There is at least one mistake.**

For example, choice (b) should be true.

**There is at least one mistake.**

For example, choice (c) should be true.

**There is at least one mistake.**

For example, choice (d) should be false.

**Your answers are correct**

**False**.**True**.**True**.**False**.

## Question 6

Which of the following is the general solution to
$\frac{{d}^{2}y}{d{x}^{2}}+10\frac{dy}{dx}+25y=0$?

In each case, $A$ and $B$ are arbitrary constants.

In each case, $A$ and $B$ are arbitrary constants.

**Not correct. Choice (a) is false.**

Note that $A{e}^{-5x}+B{e}^{-5x}=\left(A+B\right){e}^{-5x}$,
so that there is really only one arbitrary constant involved in this option.

**Your answer is correct.**

**Not correct. Choice (c) is false.**

Note
that $A{e}^{5x}+B{e}^{5x}=\left(A+B\right){e}^{5x}$,
so that there is really only one arbitrary constant involved in this option.

**Not correct. Choice (d) is false.**

The auxiliary equation
has a repeated root of $-5$.

## Question 7

Consider the differential equation $\frac{{d}^{2}y}{d{x}^{2}}-7y=0$.

Which of the following options is correct?

Which of the following options is correct?

**Not correct. Choice (a) is false.**

**Not correct. Choice (b) is false.**

**Not correct. Choice (c) is false.**

**Your answer is correct.**

## Question 8

The general solution to $\frac{{d}^{2}z}{d{t}^{2}}+6\frac{dz}{dt}+9z=0$
is

$z=At{e}^{-3t}+B{e}^{-3t}$.

Which of the following options is correct?

$z=At{e}^{-3t}+B{e}^{-3t}$.

Which of the following options is correct?

**Not correct. Choice (a) is false.**

**Not correct. Choice (b) is false.**

**Your answer is correct.**

**Not correct. Choice (d) is false.**

## Question 9

Find the particular solution to $\frac{{d}^{2}y}{d{t}^{2}}-3\frac{dy}{dt}-4y=0$
satisfying $y\left(0\right)=0$
and ${y}^{\prime}\left(0\right)=5$.

**Not correct. Choice (a) is false.**

This
function satisfies the initial conditions, but not the differential equation.

**Not correct. Choice (b) is false.**

This function satisfies the differential equation, and
$y\left(0\right)=0$, but
${y}^{\prime}\left(0\right)=-5$.

**Not correct. Choice (c) is false.**

This
functionsatisfies neither the differential equation, nor the initial conditions.

**Your answer is correct.**

## Question 10

Find a solution to $\frac{{d}^{2}x}{d{t}^{2}}+\frac{dx}{dt}-2x=0$
which satisfies $x\left(0\right)=3$
and does not tend to infinity (or minus infinity) as
$t\to \infty $.

**Not correct. Choice (a) is false.**

Try again. This function does not satisfy the differential equation.

**Not correct. Choice (b) is false.**

Try again. This
function tends to $-\infty $
as $t\to \infty $.

**Your answer is correct.**

**Not correct. Choice (d) is false.**

Try
again. This function does not satisfy the differential equation.