SAGE
Collection
Self-Hinting Language Models Enhance Reinforcement Learning • 23 items • Updated • 2
problem stringlengths 58 998 | data_source stringclasses 1
value | gt stringlengths 1 173 |
|---|---|---|
Each of the two Magellan telescopes has a diameter of $6.5 \mathrm{~m}$. In one configuration the effective focal length is $72 \mathrm{~m}$. Find the diameter of the image of a planet (in $\mathrm{cm}$ ) at this focus if the angular diameter of the planet at the time of the observation is $45^{\prime \prime}$. | minerva_math | 1.6 |
A white dwarf star has an effective temperature, $T_{e}=50,000$ degrees Kelvin, but its radius, $R_{\mathrm{WD}}$, is comparable to that of the Earth. Take $R_{\mathrm{WD}}=10^{4} \mathrm{~km}\left(10^{7} \mathrm{~m}\right.$ or $\left.10^{9} \mathrm{~cm}\right)$. Compute the luminosity (power output) of the white dwarf... | minerva_math | 4.5e33 |
Preamble: A prism is constructed from glass and has sides that form a right triangle with the other two angles equal to $45^{\circ}$. The sides are $L, L$, and $H$, where $L$ is a leg and $H$ is the hypotenuse. A parallel light beam enters side $L$ normal to the surface, passes into the glass, and then strikes $H$ inte... | minerva_math | 41.8 |
A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \mathrm{Mpc}$, what will its apparent magnitude be? | minerva_math | 20.39 |
Find the gravitational acceleration due to the Sun at the location of the Earth's orbit (i.e., at a distance of $1 \mathrm{AU}$ ). Give your answer in meters per second squared, and express it to one significant figure. | minerva_math | 0.006 |
Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\theta_w$ with respect to the surface normal.
Subproblem 0: If the index of refraction of water is $n=1.3$, find an expression for the angle of the light once it emerges from the water into the air, $\t... | minerva_math | np.arcsin(10/13) |
Find the theoretical limiting angular resolution (in arcsec) of a commercial 8-inch (diameter) optical telescope being used in the visible spectrum (at $\lambda=5000 \AA=500 \mathrm{~nm}=5 \times 10^{-5} \mathrm{~cm}=5 \times 10^{-7} \mathrm{~m}$). Answer in arcseconds to two significant figures. | minerva_math | 0.49 |
A star has a measured parallax of $0.01^{\prime \prime}$, that is, $0.01$ arcseconds. How far away is it, in parsecs? | minerva_math | 100 |
An extrasolar planet has been observed which passes in front of (i.e., transits) its parent star. If the planet is dark (i.e., contributes essentially no light of its own) and has a surface area that is $2 \%$ of that of its parent star, find the decrease in magnitude of the system during transits. | minerva_math | 0.022 |
If the Bohr energy levels scale as $Z^{2}$, where $Z$ is the atomic number of the atom (i.e., the charge on the nucleus), estimate the wavelength of a photon that results from a transition from $n=3$ to $n=2$ in Fe, which has $Z=26$. Assume that the Fe atom is completely stripped of all its electrons except for one. G... | minerva_math | 9.6 |
If the Sun's absolute magnitude is $+5$, find the luminosity of a star of magnitude $0$ in ergs/s. A useful constant: the luminosity of the sun is $3.83 \times 10^{33}$ ergs/s. | minerva_math | 3.83e35 |
Preamble: A spectrum is taken of a single star (i.e., one not in a binary). Among the observed spectral lines is one from oxygen whose rest wavelength is $5007 \AA$. The Doppler shifted oxygen line from this star is observed to be at a wavelength of $5012 \AA$. The star is also observed to have a proper motion, $\mu$, ... | minerva_math | 300 |
The differential luminosity from a star, $\Delta L$, with an approximate blackbody spectrum, is given by:
\[
\Delta L=\frac{8 \pi^{2} c^{2} R^{2}}{\lambda^{5}\left[e^{h c /(\lambda k T)}-1\right]} \Delta \lambda
\]
where $R$ is the radius of the star, $T$ is its effective surface temperature, and $\lambda$ is the wavel... | minerva_math | \frac{2\pic^{2}R^{2}}{\lambda^{5}[e^{hc/(\lambdakT)}-1]d^{2}} |
Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.
Subproblem 0: Fin... | minerva_math | 8.7e8 |
A star is at a distance from the Earth of $300 \mathrm{pc}$. Find its parallax angle, $\pi$, in arcseconds to one significant figure. | minerva_math | 0.003 |
The Sun's effective temperature, $T_{e}$, is 5800 Kelvin, and its radius is $7 \times 10^{10} \mathrm{~cm}\left(7 \times 10^{8}\right.$ m). Compute the luminosity (power output) of the Sun in erg/s. Treat the Sun as a blackbody radiator, and give your answer to one significant figure. | minerva_math | 4e33 |
Use the Bohr model of the atom to compute the wavelength of the transition from the $n=100$ to $n=99$ levels, in centimeters. [Uscful relation: the wavelength of $L \alpha$ ( $\mathrm{n}=2$ to $\mathrm{n}=1$ transition) is $1216 \AA$.] | minerva_math | 4.49 |
Preamble: A radio interferometer, operating at a wavelength of $1 \mathrm{~cm}$, consists of 100 small dishes, each $1 \mathrm{~m}$ in diameter, distributed randomly within a $1 \mathrm{~km}$ diameter circle.
What is the angular resolution of a single dish, in radians? | minerva_math | 0.01 |
Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\text {orb }}=10$ days. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is $v_{1}=20 \mathrm{~km} \mathrm{~s}^{-1}$. Star 2 is an X-... | minerva_math | 3.3e12 |
If a star cluster is made up of $10^{4}$ stars, each of whose absolute magnitude is $-5$, compute the combined apparent magnitude of the cluster if it is located at a distance of $1 \mathrm{Mpc}$. | minerva_math | 10 |
A galaxy moves directly away from us with a speed of $3000 \mathrm{~km} \mathrm{~s}^{-1}$. Find the wavelength of the $\mathrm{H} \alpha$ line observed at the Earth, in Angstroms. The rest wavelength of $\mathrm{H} \alpha$ is $6565 \AA$. Take the speed of light to be $3\times 10^8$ meters per second. | minerva_math | 6630 |
The Spitzer Space Telescope has an effective diameter of $85 \mathrm{cm}$, and a typical wavelength used for observation of $5 \mu \mathrm{m}$, or 5 microns. Based on this information, compute an estimate for the angular resolution of the Spitzer Space telescope in arcseconds. | minerva_math | 1.2 |
It has long been suspected that there is a massive black hole near the center of our Galaxy. Recently, a group of astronmers determined the parameters of a star that is orbiting the suspected black hole. The orbital period is 15 years, and the orbital radius is $0.12$ seconds of arc (as seen from the Earth). Take the d... | minerva_math | 3e6 |
Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.
Find the luminosi... | minerva_math | 7e37 |
A large ground-based telescope has an effective focal length of 10 meters. Two astronomical objects are separated by 1 arc second in the sky. How far apart will the two corresponding images be in the focal plane, in microns? | minerva_math | 50 |
The equation of state for cold (non-relativistic) matter may be approximated as:
\[
P=a \rho^{5 / 3}-b \rho^{4 / 3}
\]
where $P$ is the pressure, $\rho$ the density, and $a$ and $b$ are fixed constants. Use a dimensional analysis of the equation of hydrostatic equilibrium to estimate the ``radius-mass'' relation for pl... | minerva_math | \frac{aM^{1/3}}{GM^{2/3}+b} |
Take the total energy (potential plus thermal) of the Sun to be given by the simple expression:
\[
E \simeq-\frac{G M^{2}}{R}
\]
where $M$ and $R$ are the mass and radius, respectively. Suppose that the energy generation in the Sun were suddenly turned off and the Sun began to slowly contract. During this contraction i... | minerva_math | 7.5e7 |
Preamble: Once a star like the Sun starts to ascend the giant branch its luminosity, to a good approximation, is given by:
\[
L=\frac{10^{5} L_{\odot}}{M_{\odot}^{6}} M_{\text {core }}^{6}
\]
where the symbol $\odot$ stands for the solar value, and $M_{\text {core }}$ is the mass of the He core of the star. Further, as... | minerva_math | \frac{dM}{dt}=\frac{10^{5}L_{\odot}}{0.007c^{2}M_{\odot}^{6}}M^{6} |
A star of radius, $R$, and mass, $M$, has an atmosphere that obeys a polytropic equation of state:
\[
P=K \rho^{5 / 3} \text {, }
\]
where $P$ is the gas pressure, $\rho$ is the gas density (mass per unit volume), and $K$ is a constant throughout the atmosphere. Assume that the atmosphere is sufficiently thin (compared... | minerva_math | [P_{0}^{2/5}-\frac{2}{5}gK^{-3/5}z]^{5/2} |
An eclipsing binary consists of two stars of different radii and effective temperatures. Star 1 has radius $R_{1}$ and $T_{1}$, and Star 2 has $R_{2}=0.5 R_{1}$ and $T_{2}=2 T_{1}$. Find the change in bolometric magnitude of the binary, $\Delta m_{\text {bol }}$, when the smaller star is behind the larger star. (Consid... | minerva_math | 1.75 |
Preamble: It has been suggested that our Galaxy has a spherically symmetric dark-matter halo with a density distribution, $\rho_{\text {dark }}(r)$, given by:
\[
\rho_{\text {dark }}(r)=\rho_{0}\left(\frac{r_{0}}{r}\right)^{2},
\]
where $\rho_{0}$ and $r_{0}$ are constants, and $r$ is the radial distance from the cente... | minerva_math | \sqrt{4\piG\rho_{0}r_{0}^{2}} |
The Very Large Array (VLA) telescope has an effective diameter of $36 \mathrm{~km}$, and a typical wavelength used for observation at this facility might be $6 \mathrm{~cm}$. Based on this information, compute an estimate for the angular resolution of the VLA in arcseconds | minerva_math | 0.33 |
Subproblem 0: A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \mathrm{Mpc}$, what will its apparent magnitude be?
Solution: \[
\text { Given: } M=-7 \text { and } d=3 \mathrm{Mpc}
\]
\[
\begin{aligned}
& \text { Apparent Magnitude: } m=M+5 \log \... | minerva_math | 27.39 |
Find the distance modulus to the Andromeda galaxy (M31). Take the distance to Andromeda to be $750 \mathrm{kpc}$, and answer to three significant figures. | minerva_math | 24.4 |
The Hubble Space telescope has an effective diameter of $2.5 \mathrm{~m}$, and a typical wavelength used for observation by the Hubble might be $0.6 \mu \mathrm{m}$, or 600 nanometers (typical optical wavelength). Based on this information, compute an estimate for the angular resolution of the Hubble Space telescope in... | minerva_math | 0.05 |
Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\theta_w$ with respect to the surface normal.
If the index of refraction of water is $n=1.3$, find an expression for the angle of the light once it emerges from the water into the air, $\theta_a$, in te... | minerva_math | \arcsin{1.3\sin{\theta_w}} |
What fraction of the rest mass energy is released (in the form of radiation) when a mass $\Delta M$ is dropped from infinity onto the surface of a neutron star with $M=1 M_{\odot}$ and $R=10$ $\mathrm{km}$ ? | minerva_math | 0.15 |
Preamble: The density of stars in a particular globular star cluster is $10^{6} \mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average speed of $10 \mathrm{~km} \mathrm{sec}^{-1}$.
Find the mean free path for collisions among stars. Express your answer in centimeters, to a single... | minerva_math | 2e27 |
For a gas supported by degenerate electron pressure, the pressure is given by:
\[
P=K \rho^{5 / 3}
\]
where $K$ is a constant and $\rho$ is the mass density. If a star is totally supported by degenerate electron pressure, use a dimensional analysis of the equation of hydrostatic equilibrium:
\[
\frac{d P}{d r}=-g \rho
... | minerva_math | -1./3 |
A galaxy moves directly away from us with speed $v$, and the wavelength of its $\mathrm{H} \alpha$ line is observed to be $6784 \AA$. The rest wavelength of $\mathrm{H} \alpha$ is $6565 \AA$. Find $v/c$. | minerva_math | 0.033 |
A candle has a power in the visual band of roughly $3$ Watts. When this candle is placed at a distance of $3 \mathrm{~km}$ it has the same apparent brightness as a certain star. Assume that this star has the same luminosity as the Sun in the visual band $\left(\sim 10^{26}\right.$ Watts $)$. How far away is the star (i... | minerva_math | 0.5613 |
Preamble: A galaxy is found to have a rotation curve, $v(r)$, given by
\[
v(r)=\frac{\left(\frac{r}{r_{0}}\right)}{\left(1+\frac{r}{r_{0}}\right)^{3 / 2}} v_{0}
\]
where $r$ is the radial distance from the center of the galaxy, $r_{0}$ is a constant with the dimension of length, and $v_{0}$ is another constant with the... | minerva_math | \frac{v_{0}}{r_{0}}\frac{1}{(1+r/r_{0})^{3/2}} |
Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\text {orb }}=10$ days. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is $v_{1}=20 \mathrm{~km} \mathrm{~s}^{-1}$. Star 2 is an X-... | minerva_math | 2.75e11 |
Preamble: The density of stars in a particular globular star cluster is $10^{6} \mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average speed of $10 \mathrm{~km} \mathrm{sec}^{-1}$.
Subproblem 0: Find the mean free path for collisions among stars. Express your answer in centimeter... | minerva_math | 6e13 |
Preamble: A radio interferometer, operating at a wavelength of $1 \mathrm{~cm}$, consists of 100 small dishes, each $1 \mathrm{~m}$ in diameter, distributed randomly within a $1 \mathrm{~km}$ diameter circle.
Subproblem 0: What is the angular resolution of a single dish, in radians?
Solution: The angular resolution... | minerva_math | 1e-5 |
If a star cluster is made up of $10^{6}$ stars whose absolute magnitude is the same as that of the Sun (+5), compute the combined magnitude of the cluster if it is located at a distance of $10 \mathrm{pc}$. | minerva_math | -10 |
A certain red giant has a radius that is 500 times that of the Sun, and a temperature that is $1 / 2$ that of the Sun's temperature. Find its bolometric (total) luminosity in units of the bolometric luminosity of the Sun. | minerva_math | 15625 |
Suppose air molecules have a collision cross section of $10^{-16} \mathrm{~cm}^{2}$. If the (number) density of air molecules is $10^{19} \mathrm{~cm}^{-3}$, what is the collision mean free path in cm? Answer to one significant figure. | minerva_math | 1e-3 |
Two stars have the same surface temperature. Star 1 has a radius that is $2.5$ times larger than the radius of star 2. Star 1 is ten times farther away than star 2. What is the absolute value of the difference in apparent magnitude between the two stars, rounded to the nearest integer? | minerva_math | 3 |
What is the slope of a $\log N(>F)$ vs. $\log F$ curve for a homogeneous distribution of objects, each of luminosity, $L$, where $F$ is the flux at the observer, and $N$ is the number of objects observed per square degree on the sky? | minerva_math | -3./2 |
Preamble: Comparison of Radio and Optical Telescopes.
The Very Large Array (VLA) is used to make an interferometric map of the Orion Nebula at a wavelength of $10 \mathrm{~cm}$. What is the best angular resolution of the radio image that can be produced, in radians? Note that the maximum separation of two antennae in ... | minerva_math | 2.7778e-6 |
A globular cluster has $10^{6}$ stars each of apparent magnitude $+8$. What is the combined apparent magnitude of the entire cluster? | minerva_math | -7 |
Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.
Subproblem 0: Fin... | minerva_math | 48 |
A Boolean function $F(A, B)$ is said to be universal if any arbitrary boolean function can be constructed by using nested $F(A, B)$ functions. A universal function is useful, since using it we can build any function we wish out of a single part. For example, when implementing boolean logic on a computer chip a universa... | minerva_math | 16 |
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 ... | minerva_math | 0.192 |
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 ... | minerva_math | 0.5978 |
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... | minerva_math | 11000 |
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 fe... | minerva_math | 0.53 |
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... | minerva_math | 15000 |
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.
Gi... | minerva_math | 1538 |
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 the... | minerva_math | 4.4e7 |
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 capacita... | minerva_math | I(0) |
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 r... | minerva_math | y^{\prime}+ry-rx(t)=0 |
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 pro... | minerva_math | cx^{-2}-\frac{\cos(2x)}{2x^{2}} |
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 ... | minerva_math | \frac{dx}{dt}=kx-a |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the magnitude of $z^2$? | minerva_math | 4 |
In the polar representation $(r, \theta)$ of the complex number $z=1+\sqrt{3} i$, what is $r$? | minerva_math | 2 |
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. | minerva_math | 1+\sqrt{3}i |
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 ... | minerva_math | 0 |
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}... | minerva_math | -0.875 |
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 $b$. | minerva_math | \sqrt{2}\cos(2t-\pi/4) |
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 $\dot{x}(0)=1$. | minerva_math | \frac{1}{2a}(\exp{a*t}-\exp{-a*t}) |
Find a solution to the differential equation $\ddot{x}+\omega^{2} x=0$ satisfying the initial conditions $x(0)=x_{0}$ and $\dot{x}(0)=\dot{x}_{0}$. | minerva_math | x_{0}\cos(\omegat)+\dot{x}_{0}\sin(\omegat)/\omega |
Find the complex number $a+b i$ with the smallest possible positive $b$ such that $e^{a+b i}=1+\sqrt{3} i$. | minerva_math | \ln2+i\pi/3 |
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^{... | minerva_math | e^{t}/3 |
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... | minerva_math | 2 |
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 ... | minerva_math | \frac{\cos(2t)}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\cos(t)+C_{4}\sin(t) |
For $\omega \geq 0$, find $A$ such that $A \cos (\omega t)$ is a solution of $\ddot{x}+4 x=\cos (\omega t)$. | minerva_math | \frac{1}{4-\omega^{2}} |
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. | minerva_math | \frac{\cos(2t)+\sin(2t)}{4} |
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 $\ddot{x}+4 x=\sin (3 t)$. | minerva_math | -0.2 |
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_... | minerva_math | x/2-1/4+ce^{-2x} |
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 $\frac{e^{-2 t}}{p(-2)}=\boxed{\frac{e^{-2 t}}{15}}$... | minerva_math | \frac{e^{-2t}}{15}+C_{1}e^{t}+C_{2}e^{-t}+C_{3}\cos(t)+C_{4}\sin(t) |
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}+\omeg... | minerva_math | s^{2}+\omega^{2} |
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 $b$. | minerva_math | 2\cos(\pit+\pi/3) |
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 $\omega \neq 0$. | minerva_math | 2\pi/\omega |
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 ... | minerva_math | \frac{4A}{3+3i}e^{it} |
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 capacita... | minerva_math | I(0)e^{-\frac{t}{RC}} |
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^{... | minerva_math | \frac{e^{2it}}{(2+2i)} |
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_... | minerva_math | \frac{2-\omega^{2}-\omegai/4}{\omega^{4}-\frac{63}{16}\omega^{2}+4} |
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 so... | minerva_math | x^{\prime}+\frac{r}{V}x-rc=0 |
Find the polynomial solution of $\ddot{x}-x=t^{2}+t+1$, solving for $x(t)$. | minerva_math | -t^2-t-3 |
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... | minerva_math | -8 |
Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$. | minerva_math | \frac{\cos(2t)}{15} |
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... | minerva_math | -2+2\sqrt{3}i |
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... | minerva_math | \frac{t}{4}\sin(2t) |
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 $\dot{x}(0)=0$. | minerva_math | \frac{1}{2}(\exp{a*t}+\exp{-a*t}) |
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. | minerva_math | \frac{e^{t}}{3}+ce^{-2t} |
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. | minerva_math | (\frac{t^{2}}{2}-t+C)e^{-t} |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the real part of $z^2$? | minerva_math | -2 |
Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$. | minerva_math | \frac{e^{-2t}}{15} |
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