## MATH1003 Quizzes

Quiz 11: Second Order Linear Differential Equations
Question 1 Questions
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!
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!
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)
 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) $y=A{e}^{-2x}+B{e}^{-5x}$

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)
 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) $y=A{e}^{\left(-2+2\sqrt{2}\right)t}+B{e}^{\left(-2-2\sqrt{2}\right)t}$

Choice (a) is incorrect
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Choice (b) is incorrect
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Choice (c) is incorrect
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
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)
 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.
Correct!
1. False
2. True
3. True
4. 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)
 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) $y=Ax{e}^{5x}+B{e}^{5x}$

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)
 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 $-\sqrt{7}$.

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)
 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 $B$.

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)
 a) $y={e}^{t}-{e}^{-4t}$ b) $y={e}^{-t}-{e}^{4t}$ c) $y={e}^{-4t}-{e}^{t}$ d) $y={e}^{4t}-{e}^{-t}$

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

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.