ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Unfortunately, a mutant gene can turn box people into triangles late in life. A laboratory test has been developed which can spot the gene early so that the dreaded triangle transformation can be prevented by medications. This test is 95 percent accurate at spotting the gene when it is there. However, the test gives a "false positive" $0.4$ percent of the time, falsely indicating that a healthy box person has the mutant gene. If $0.1$ percent (be careful - that's one-tenth of one percent) of the box people have the mutant gene, what's the probability that a box person actually has the mutant gene if the test indicates that he or she does?
Answer: 0.192
Problem: Buzz, the hot new dining spot on campus, emphasizes simplicity. It only has two items on the menu, burgers and zucchini. Customers make a choice as they enter (they are not allowed to order both), and inform the cooks in the back room by shouting out either "B" or "Z". Unfortunately the two letters sound similar so $8 \%$ of the time the cooks misinterpret what was said. The marketing experts who designed the restaurant guess that $90 \%$ of the orders will be for burgers and $10 \%$ for zucchini.
The cooks can hear one order per second. The customers arrive at the rate of one per second. One of the chefs says that this system will never work because customers can only send one bit per second, the rate at which orders can be accepted, so you could barely keep up even if there were no noise in the channel. You are hired as an outside consultant to deal with the problem.
What is the channel capacity $\mathrm{C}$ of this communication channel in bits per second?
Answer: 0.5978
Problem: Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.
$\begin{array}{ll}\text { Total vegetative biomass } & 80,000 \mathrm{kcal} \mathrm{m}^{-2} \\ \text { Detritus and organic matter in soil } & 120,000 \mathrm{kcal } \mathrm{m}^{-2} \\ \text { Total Gross Primary Productivity } & 20,000 \mathrm{kcal } \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Plant Respiration } & 5,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Community Respiration } & 9,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1}\end{array}$

Subproblem 0: What is the net primary productivity of the forest?


Solution: NPP $=$ GPP $-R_{A}=20,000-5,000=\boxed{15000} \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1}$

Final answer: The final answer is 15000. I hope it is correct.

Subproblem 1: What is the net community production?
Answer: 11000
Problem: Preamble: A population of 100 ferrets is introduced to a large island in the beginning of 1990 . Ferrets have an intrinsic growth rate, $r_{\max }$ of $1.3 \mathrm{yr}^{-1}$.

Subproblem 0: Assuming unlimited resources-i.e., there are enough resources on this island to last the ferrets for hundreds of years-how many ferrets will there be on the island in the year 2000? (Show your work!)


Solution: $N_o = 100$ (in 1990)
\\
$N = ?$ (in 2000)
\\
$t = 10$ yr
\\
$r = 1.3 \text{yr}^{-1}$
\\
$N = N_{o}e^{rt} = 100*e^{(1.3/\text{yr})(10 \text{yr})} = 4.4 x 10^7$ ferrets
\\
There will be \boxed{4.4e7} ferrets on the island in the year 2000. 

Final answer: The final answer is 4.4e7. I hope it is correct.

Subproblem 1: What is the doubling time of the ferret population? (Show your work!)
Answer: 0.53
Problem: Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.
$\begin{array}{ll}\text { Total vegetative biomass } & 80,000 \mathrm{kcal} \mathrm{m}^{-2} \\ \text { Detritus and organic matter in soil } & 120,000 \mathrm{kcal } \mathrm{m}^{-2} \\ \text { Total Gross Primary Productivity } & 20,000 \mathrm{kcal } \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Plant Respiration } & 5,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Community Respiration } & 9,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1}\end{array}$

What is the net primary productivity of the forest?
Answer: 15000
Problem: Preamble: The Peak District Moorlands in the United Kingdom store 20 million tonnes of carbon, almost half of the carbon stored in the soils of the entire United Kingdom (the Moorlands are only $8 \%$ of the land area). In pristine condition, these peatlands can store an additional 13,000 tonnes of carbon per year.

Given this rate of productivity, how long did it take for the Peatlands to sequester this much carbon?
Answer: 1538
Problem: Preamble: A population of 100 ferrets is introduced to a large island in the beginning of 1990 . Ferrets have an intrinsic growth rate, $r_{\max }$ of $1.3 \mathrm{yr}^{-1}$.

Assuming unlimited resources-i.e., there are enough resources on this island to last the ferrets for hundreds of years-how many ferrets will there be on the island in the year 2000? (Show your work!)
Answer: 4.4e7
Problem: Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circuit and by $V(t)$ the voltage increase across the voltage source, at time $t$. Denote by $R$ the resistance of the resistor and $C$ the capacitance of the capacitor (in units which we will not specify)-both positive numbers. Then
\[
R \dot{I}+\frac{1}{C} I=\dot{V}
\]

Subproblem 0: Suppose that $V$ is constant, $V(t)=V_{0}$. Solve for $I(t)$, with initial condition $I(0)$.


Solution: When $V$ is constant, the equation becomes $R \dot{I}+\frac{1}{C} I=0$, which is separable. Solving gives us
\[
I(t)=\boxed{I(0) e^{-\frac{t}{R C}}
}\]. 

Final answer: The final answer is I(0) e^{-\frac{t}{R C}}
. I hope it is correct.

Subproblem 1: It is common to write the solution to the previous subproblem in the form $c e^{-t / \tau}$. What is $c$ in this case?
Answer: I(0)
Problem: Consider the following "mixing problem." A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \mathrm{gm} /$ liter is added at the rate of $r$ liters/minute. A mixer keeps the salt essentially uniformly distributed in the tank. A pipe lets solution out of the tank at the same rate of $r$ liters/minute. The differential equation for the amount of salt in the tank is given by 
\[
x^{\prime}+\frac{r}{V} x-r c=0 .
\]
Suppose that the out-flow from this tank leads into another tank, also of volume 1 , and that at time $t=1$ the water in it has no salt in it. Again there is a mixer and an outflow. Write down a differential equation for the amount of salt in this second tank, as a function of time, assuming the amount of salt in the second tank at moment $t$ is given by $y(t)$, and the amount of salt in the first tank at moment $t$ is given by $x(t)$.
Answer: y^{\prime}+ry-rx(t)=0
Problem: Find the general solution of $x^{2} y^{\prime}+2 x y=\sin (2 x)$, solving for $y$. Note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where $x_{h}$ is a nonzero solution of the homogeneous equation $\dot{x}+p x=0$. Additionally, note that the left hand side is the derivative of a product.
Answer: cx^{-2}-\frac{\cos(2x)}{2x^{2}}
Problem: An African government is trying to come up with good policy regarding the hunting of oryx. They are using the following model: the oryx population has a natural growth rate of $k$, and we suppose a constant harvesting rate of $a$ oryxes per year.
Write down an ordinary differential equation describing the evolution of the oryx population given the dynamics above, using $x(t)$ to denote the oryx population (the number of individual oryx(es)) at time $t$, measured in years.
Answer: \frac{dx}{dt}=kx-a
Problem: If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the magnitude of $z^2$?
Answer: 4
Problem: In the polar representation $(r, \theta)$ of the complex number $z=1+\sqrt{3} i$, what is $r$?
Answer: 2
Problem: Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers.
Answer: 1+\sqrt{3}i
Problem: Subproblem 0: Find the general solution of the differential equation $y^{\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such that $u \dot{x}+u p x=\frac{d}{d t}(u x)$ is an integrating factor. Additionally, note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where $x_{h}$ is a nonzero solution of the homogeneous equation $\dot{x}+p x=0$.


Solution: In standard form, $y^{\prime}+2 y=x$, so $u=C e^{2 x}$. Then $y=u^{-1} \int u x d x=e^{-2 x} \int x e^{2 x} d x$. Integrating by parts yields $\int x e^{2 x} d x=$ $\frac{x}{2} e^{2 x}-\frac{1}{2} \int e^{2 x} d x=\frac{x}{2} e^{2 x}-\frac{1}{4} e^{2 x}+c$. Therefore, $y=\boxed{x / 2-1 / 4+c e^{-2 x}}$.

Final answer: The final answer is x / 2-1 / 4+c e^{-2 x}. I hope it is correct.

Subproblem 1: For what value of $c$ does the straight line solution occur?
Answer: 0
Problem: Preamble: The following subproblems relate to applying Euler's Method (a first-order numerical procedure for solving ordinary differential equations with a given initial value) onto $y^{\prime}=y^{2}-x^{2}=F(x, y)$ at $y(0)=-1$, with $h=0.5$. Recall the notation \[x_{0}=0, y_{0}=-1, x_{n+1}=x_{h}+h, y_{n+1}=y_{n}+m_{n} h, m_{n}=F\left(x_{n}, y_{n}\right)\]. 

Use Euler's method to estimate the value at $x=1.5$.
Answer: -0.875
Problem: Rewrite the function $f(t) = \cos (2 t)+\sin (2 t)$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$.
Answer: \sqrt{2}\cos(2t-\pi/4)
Problem: Given the ordinary differential equation $\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 0$ and $\dot{x}(0)=1$.
Answer: \frac{1}{2a}(\exp{a*t}-\exp{-a*t})
Problem: Find a solution to the differential equation $\ddot{x}+\omega^{2} x=0$ satisfying the initial conditions $x(0)=x_{0}$ and $\dot{x}(0)=\dot{x}_{0}$.
Answer: x_{0}\cos(\omegat)+\dot{x}_{0}\sin(\omegat)/\omega
Problem: Find the complex number $a+b i$ with the smallest possible positive $b$ such that $e^{a+b i}=1+\sqrt{3} i$.
Answer: \ln2+i\pi/3
Problem: Subproblem 0: Find the general solution of the differential equation $\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur.


Solution: We can use integrating factors to get $(u x)^{\prime}=u e^{t}$ for $u=e^{2 t}$. Integrating yields $e^{2 t} x=e^{3 t} / 3+c$, or $x=\boxed{\frac{e^{t}} {3}+c e^{-2 t}}$. 

Final answer: The final answer is \frac{e^{t}} {3}+c e^{-2 t}. I hope it is correct.

Subproblem 1: Find a solution of the differential equation $\dot{x}+2 x=e^{t}$ of the form $w e^{t}$, where $w$ is a constant (which you should find).
Answer: e^{t}/3
Problem: Subproblem 0: For $\omega \geq 0$, find $A$ such that $A \cos (\omega t)$ is a solution of $\ddot{x}+4 x=\cos (\omega t)$.


Solution: If $x=A \cos (\omega t)$, then taking derivatives gives us $\ddot{x}=-\omega^{2} A \cos (\omega t)$, and $\ddot{x}+4 x=\left(4-\omega^{2}\right) A \cos (\omega t)$. Then $A=\boxed{\frac{1}{4-\omega^{2}}}$. 

Final answer: The final answer is \frac{1}{4-\omega^{2}}. I hope it is correct.

Subproblem 1: For what value of $\omega$ does resonance occur?
Answer: 2
Problem: Subproblem 0: Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$.


Solution: We choose an exponential input function whose real part is $\cos (2 t)$, namely $e^{2 i t}$. Since $p(s)=s^{4}-1$ and $p(2 i)=15 \neq 0$, the exponential response formula yields the solution $\frac{e^{2 i t}}{15}$. A sinusoidal solution to the original equation is given by the real part: $\boxed{\frac{\cos (2 t)}{15}}$. 

Final answer: The final answer is \frac{\cos (2 t)}{15}. I hope it is correct.

Subproblem 1: Find the general solution to $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$, denoting constants as $C_{1}, C_{2}, C_{3}, C_{4}$.
Answer: \frac{\cos(2t)}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\cos(t)+C_{4}\sin(t)
Problem: For $\omega \geq 0$, find $A$ such that $A \cos (\omega t)$ is a solution of $\ddot{x}+4 x=\cos (\omega t)$.
Answer: \frac{1}{4-\omega^{2}}
Problem: Find a solution to $\dot{x}+2 x=\cos (2 t)$ in the form $k_0\left[f(k_1t) + g(k_2t)\right]$, where $f, g$ are trigonometric functions.  Do not include homogeneous solutions to this ODE in your solution.
Answer: \frac{\cos(2t)+\sin(2t)}{4}
Problem: Preamble: The following subproblems refer to the differential equation. $\ddot{x}+4 x=\sin (3 t)$

Find $A$ so that $A \sin (3 t)$ is a solution of $\ddot{x}+4 x=\sin (3 t)$.
Answer: -0.2
Problem: Find the general solution of the differential equation $y^{\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such that $u \dot{x}+u p x=\frac{d}{d t}(u x)$ is an integrating factor. Additionally, note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where $x_{h}$ is a nonzero solution of the homogeneous equation $\dot{x}+p x=0$.
Answer: x/2-1/4+ce^{-2x}
Problem: Subproblem 0: Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$.


Solution: The characteristic polynomial of the homogeneous equation is given by $p(s)=$ $s^{4}-1$. Since $p(-2)=15 \neq 0$, the exponential response formula gives the solution $\frac{e^{-2 t}}{p(-2)}=\boxed{\frac{e^{-2 t}}{15}}$.

Final answer: The final answer is \frac{e^{-2 t}}{15}. I hope it is correct.

Subproblem 1: Find the general solution to $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$, denoting constants as $C_{1}, C_{2}, C_{3}, C_{4}$.
Answer: \frac{e^{-2t}}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\cos(t)+C_{4}\sin(t)
Problem: Preamble: Consider the differential equation $\ddot{x}+\omega^{2} x=0$. \\

A differential equation $m \ddot{x}+b \dot{x}+k x=0$ (where $m, b$, and $k$ are real constants, and $m \neq 0$ ) has corresponding characteristic polynomial $p(s)=m s^{2}+b s+k$.\\
What is the characteristic polynomial $p(s)$ of $\ddot{x}+\omega^{2} x=0$?
Answer: s^{2}+\omega^{2}
Problem: Rewrite the function $\cos (\pi t)-\sqrt{3} \sin (\pi t)$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$.
Answer: 2\cos(\pit+\pi/3)
Problem: Preamble: The following subproblems refer to the damped sinusoid $x(t)=A e^{-a t} \cos (\omega t)$.

What is the spacing between successive maxima of $x(t)$? Assume that $\omega \neq 0$.
Answer: 2\pi/\omega
Problem: Preamble: The following subproblems refer to a spring/mass/dashpot system driven through the spring modeled by the equation $m \ddot{x}+b \dot{x}+k x=k y$. Here $x$ measures the position of the mass, $y$ measures the position of the other end of the spring, and $x=y$ when the spring is relaxed.

In this system, regard $y(t)$ as the input signal and $x(t)$ as the system response. Take $m=1, b=3, k=4, y(t)=A \cos t$. Replace the input signal by a complex exponential $y_{c x}(t)$ of which it is the real part, and compute the exponential ("steady state") system response $z_p(t)$; leave your answer in terms of complex exponentials, i.e. do not take the real part.
Answer: \frac{4A}{3+3i}e^{it}
Problem: Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circuit and by $V(t)$ the voltage increase across the voltage source, at time $t$. Denote by $R$ the resistance of the resistor and $C$ the capacitance of the capacitor (in units which we will not specify)-both positive numbers. Then
\[
R \dot{I}+\frac{1}{C} I=\dot{V}
\]

Suppose that $V$ is constant, $V(t)=V_{0}$. Solve for $I(t)$, with initial condition $I(0)$.
Answer: I(0)e^{-\frac{t}{RC}}
Problem: Subproblem 0: Find the general (complex-valued) solution of the differential equation $\dot{z}+2 z=e^{2 i t}$, using $C$ to stand for any complex-valued integration constants which may arise.


Solution: Using integrating factors, we get $e^{2 t} z=e^{(2+2 i) t} /(2+2 i)+C$, or $z=\boxed{\frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}}$, where $C$ is any complex number.

Final answer: The final answer is \frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}. I hope it is correct.

Subproblem 1: Find a solution of the differential equation $\dot{z}+2 z=e^{2 i t}$ in the form $w e^{t}$, where $w$ is a constant (which you should find).
Answer: \frac{e^{2it}}{(2+2i)}
Problem: Preamble: The following subproblems consider a second order mass/spring/dashpot system driven by a force $F_{\text {ext }}$ acting directly on the mass: $m \ddot{x}+b \dot{x}+k x=F_{\text {ext }}$. So the input signal is $F_{\text {ext }}$ and the system response is $x$. We're interested in sinusoidal input signal, $F_{\text {ext }}(t)=A \cos (\omega t)$, and in the steady state, sinusoidal system response, $x_{p}(t)=g A \cos (\omega t-\phi)$. Here $g$ is the gain of the system and $\phi$ is the phase lag. Both depend upon $\omega$, and we will consider how that is the case. \\
Take $A=1$, so the amplitude of the system response equals the gain, and take $m=1, b=\frac{1}{4}$, and $k=2$.\\

Compute the complex gain $H(\omega)$ of this system. (This means: make the complex replacement $F_{\mathrm{cx}}=e^{i \omega t}$, and express the exponential system response $z_{p}$ as a complex multiple of $F_{\mathrm{cx}}, i.e. z_{p}=H(\omega) F_{\mathrm{cx}}$).
Answer: \frac{2-\omega^{2}-\omegai/4}{\omega^{4}-\frac{63}{16}\omega^{2}+4}
Problem: Preamble: The following subproblems refer to the following "mixing problem": A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \mathrm{gm} /$ liter is added at the rate of $r$ liters/minute. A mixer keeps the salt essentially uniformly distributed in the tank. A pipe lets solution out of the tank at the same rate of $r$ liters/minute. 

Write down the differential equation for the amount of salt in the tank in standard linear form. [Not the concentration!] Use the notation $x(t)$ for the number of grams of salt in the tank at time $t$.
Answer: x^{\prime}+\frac{r}{V}x-rc=0
Problem: Find the polynomial solution of $\ddot{x}-x=t^{2}+t+1$, solving for $x(t)$.
Answer: -t^2-t-3
Problem: Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.

Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.

Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.


Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-2+2 \sqrt{3} i}$.

Final answer: The final answer is -2+2 \sqrt{3} i. I hope it is correct.

Subproblem 2: Rewrite $e^{3(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Answer: -8
Problem: Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$.
Answer: \frac{\cos(2t)}{15}
Problem: Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.

Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.

Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Answer: -2+2\sqrt{3}i
Problem: Find a solution of $\ddot{x}+4 x=\cos (2 t)$, solving for $x(t)$, by using the ERF on a complex replacement. The ERF (Exponential Response Formula) states that a solution to $p(D) x=A e^{r t}$ is given by $x_{p}=A \frac{e^{r t}}{p(r)}$, as long as $\left.p (r\right) \neq 0$). The ERF with resonance assumes that $p(r)=0$ and states that a solution to $p(D) x=A e^{r t}$ is given by $x_{p}=A \frac{t e^{r t}}{p^{\prime}(r)}$, as long as $\left.p^{\prime} ( r\right) \neq 0$.
Answer: \frac{t}{4}\sin(2t)
Problem: Given the ordinary differential equation $\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 1$ and $\dot{x}(0)=0$.
Answer: \frac{1}{2}(\exp{a*t}+\exp{-a*t})
Problem: Find the general solution of the differential equation $\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur.
Answer: \frac{e^{t}}{3}+ce^{-2t}
Problem: Find a solution of $\ddot{x}+3 \dot{x}+2 x=t e^{-t}$ in the form $x(t)=u(t) e^{-t}$ for some function $u(t)$.  Use $C$ for an arbitrary constant, should it arise.
Answer: (\frac{t^{2}}{2}-t+C)e^{-t}
Problem: If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the real part of $z^2$?
Answer: -2
Problem: Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$.
Answer: \frac{e^{-2t}}{15}
Problem: Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \cos (3 t)$, which we will assume is a solution of the differential equation $m \ddot{x}+b \dot{x}+k x=0$. 

What is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$.
Answer: m
Problem: Preamble: The following subproblems refer to the differential equation. $\ddot{x}+4 x=\sin (3 t)$

Subproblem 0: Find $A$ so that $A \sin (3 t)$ is a solution of $\ddot{x}+4 x=\sin (3 t)$.


Solution: We can find this by brute force. If $x=A \sin (3 t)$, then $\ddot{x}=-9 A \sin (3 t)$, so $\ddot{x}+4 x=-5 A \sin (3 t)$. Therefore, when $A=\boxed{-0.2}, x_{p}(t)=-\sin (3 t) / 5$ is a solution of the given equation.

Final answer: The final answer is -0.2. I hope it is correct.

Subproblem 1: What is the general solution, in the form $f_0(t) + C_1f_1(t) + C_2f_2(t)$, where $C_1, C_2$ denote arbitrary constants?
Answer: -\sin(3t)/5+C_{1}\sin(2t)+C_{2}\cos(2t)
Problem: What is the smallest possible positive $k$ such that all functions $x(t)=A \cos (\omega t-\phi)$---where $\phi$ is an odd multiple of $k$---satisfy $x(0)=0$? \\
Answer: \frac{\pi}{2}
Problem: Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\

What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$?
Answer: s^{2}+bs+1
Problem: Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \cos (3 t)$, which we will assume is a solution of the differential equation $m \ddot{x}+b \dot{x}+k x=0$. 

Subproblem 0: What is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$.


Solution: We can write $e^{-t / 2} \cos (3 t)=\operatorname{Re} e^{(-1 / 2 \pm 3 i) t}$, so $p(s)=m s^{2}+b s+k$ has solutions $-\frac{1}{2} \pm 3 i$. This means $p(s)=m(s+1 / 2-3 i)(s+1 / 2+3 i)=m\left(s^{2}+s+\frac{37}{4}\right)$. Then $b=\boxed{m}$, 

Final answer: The final answer is m. I hope it is correct.

Subproblem 1: What is $k$ in terms of $m$? Write $k$ as a constant times a function of $m$.
Answer: \frac{37}{4}m
Problem: Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$. 

Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.

Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.

Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.


Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-2+2 \sqrt{3} i}$.

Final answer: The final answer is -2+2 \sqrt{3} i. I hope it is correct.

Subproblem 2: Rewrite $e^{3(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers. 


Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-8}$.

Final answer: The final answer is -8. I hope it is correct.

Subproblem 3: Rewrite $e^{4(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Answer: -8-8\sqrt{3}i
Problem: Rewrite the function $\operatorname{Re} \frac{e^{i t}}{2+2 i}$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$.
Answer: \frac{\sqrt{2}}{4}\cos(t-\pi/4)