Quiz 11: Second Order Linear Differential Equations

Question

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 $k$?

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 $k$?

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.
 a) $y=A{e}^{2x}+B{e}^{-5x}$ b) $y=A{e}^{-2x}+B{e}^{5x}$ c) $y=A{e}^{2x}+B{e}^{5x}$ d) $y=A{e}^{-2x}+B{e}^{-5x}$

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.
 a) $y=A{e}^{-2t}+Bt{e}^{-2t}$ b) $y=A{e}^{2t}+B{e}^{-2t}$ c) $y=A{e}^{2t}+B{e}^{-6t}$ d) $y=A{e}^{\left(-2+2\sqrt{2}\right)t}+B{e}^{\left(-2-2\sqrt{2}\right)t}$

Not correct. Choice (a) is false.
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Not correct. Choice (b) is false.
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Not correct. Choice (c) is false.
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.

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.
 a) $x=A{e}^{5t}+B{e}^{-t}$ b) $x=A{e}^{\left(2+3i\right)t}+B{e}^{\left(2-3i\right)t}$ c) $x={e}^{2x}\left(Acos3x+Bsin3x\right)$ d) $x={e}^{3x}\left(Acos2x+Bsin2x\right)$

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.
1. False.
2. True.
3. True.
4. 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.
 a) $y=A{e}^{-5x}+B{e}^{-5x}$ b) $y=Ax{e}^{-5x}+B{e}^{-5x}$ c) $y=A{e}^{5x}+B{e}^{5x}$ d) $y=Ax{e}^{5x}+B{e}^{5x}$

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.
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?
 a) The roots of the auxiliary equation are 0 and 7. b) There is no auxiliary equation for a differential equation of this type. c) The auxiliary equation has a repeated root of $\sqrt{7}$. d) The roots of the auxiliary equation $\sqrt{7}$ and $-\sqrt{7}$.

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.

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?
 a) As $t\to \infty$, $z\to A$ for any value of $B$. b) The behaviour of $z$ as $t\to \infty$ depends on the values of $A$ and $B$. c) As $t\to \infty$, $z\to 0$ for any values of $A$ and $B$. d) As $t\to \infty$, $z\to \infty$ for any values of $A$ and $B$.

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
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 ${y}^{\prime }\left(0\right)=5$.
 a) $y={e}^{t}-{e}^{-4t}$ b) $y={e}^{-t}-{e}^{4t}$ c) $y={e}^{-4t}-{e}^{t}$ d) $y={e}^{4t}-{e}^{-t}$

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.

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 $t\to \infty$.
 a) $x=3{e}^{-t}$ b) $x=4{e}^{-2t}-{e}^{t}$ c) $x=3{e}^{-2t}$ d) $x=2{e}^{-2t}+{e}^{-t}$

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 $t\to \infty$.