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 $k$?

*Correct!*

*Incorrect.*

*Please try again.*

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 $k$?

*Correct!*

*Incorrect.*

*Please try again.*

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. Exactly one option must be correct)

In each case, $A$ and $B$ are arbitrary constants. Exactly one option must be correct)

*Choice (a) is correct!*

*Choice (b) is incorrect*

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

*Choice (c) is incorrect*

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

*Choice (d) is incorrect*

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

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. Exactly one option must be correct)

In each case, $A$ and $B$ are arbitrary constants. Exactly one option must be correct)

*Choice (a) is incorrect*

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

*Choice (b) is incorrect*

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

*Choice (c) is incorrect*

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

*Choice (d) is correct!*

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. (Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be True.

For example, choice (d) should be False.

In each case, $A$ and $B$ are arbitrary constants. More than one option may be correct. (Zero or more options can 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.

*Correct!*

*False**True**True**False*

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. Exactly one option must be correct)

In each case, $A$ and $B$ are arbitrary constants. Exactly one option must be correct)

*Choice (a) is incorrect*

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.

*Choice (b) is correct!*

*Choice (c) is incorrect*

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.

*Choice (d) is incorrect*

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

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

Which of the following options is correct? Exactly one option must be correct)

Which of the following options is correct? Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*

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? Exactly one option must be correct)

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

Which of the following options is correct? Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*

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
${y}^{\prime}\left(0\right)=5$. Exactly one option
must be correct)

*Choice (a) is incorrect*

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

*Choice (b) is incorrect*

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

*Choice (c) is incorrect*

This
function satisfies neither the differential equation, nor the initial conditions.

*Choice (d) is correct!*

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 $. Exactly one option
must be correct)

*Choice (a) is incorrect*

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

*Choice (b) is incorrect*

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

*Choice (c) is correct!*

*Choice (d) is incorrect*

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