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stringlengths 14
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stringlengths 14
1.32k
| answer
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stringlengths 1
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college_math.A_First_Course_in_Linear_Algebra
|
exercise.7.2.1
|
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
5 & -18 & -32 \\
0 & 5 & 4 \\
2 & -5 & -11
\end{array}\right]
$$
One eigenvalue is 1. Diagonalize if possible.
|
The eigenvalues are $-1,-1,1$. The eigenvectors corresponding to the eigenvalues are:
$$
\left\{\left[\begin{array}{c}
10 \\
-2 \\
3
\end{array}\right]\right\} \leftrightarrow-1,\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \leftrightarrow 1
$$
Therefore this matrix is not diagonalizable.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
The eigenvalues are $-1,-1,1$. The eigenvectors corresponding to the eigenvalues are:
$$
\left\{\left[\begin{array}{c}
10 \\
-2 \\
3
\end{array}\right]\right\} \leftrightarrow-1,\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \leftrightarrow 1
$$
Therefore this matrix is not diagonalizable.
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.1.2.43
|
Suppose the coefficient matrix of a system of $n$ equations with $n$ variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.
|
Yes. It has a unique solution.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
Yes. It has a unique solution.
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.2.1.13
|
Let $X=\left[\begin{array}{lll}-1 & -1 & 1\end{array}\right]$ and $Y=\left[\begin{array}{lll}0 & 1 & 2\end{array}\right]$. Find $X^{T} Y$ and $X Y^{T}$ if possible.
|
$X^{T} Y=\left[\begin{array}{rrr}0 & -1 & -2 \\ 0 & -1 & -2 \\ 0 & 1 & 2\end{array}\right], X Y^{T}=1$
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
X^{T} Y=\left[\begin{array}{rrr}0 & -1 & -2 \\ 0 & -1 & -2 \\ 0 & 1 & 2\end{array}\right], X Y^{T}=1
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.2.1.42
|
Let
$$
A=\left[\begin{array}{lll}
1 & 2 & 1 \\
2 & 1 & 4 \\
4 & 5 & 10
\end{array}\right]
$$
Find $A^{-1}$ if possible. If $A^{-1}$ does not exist, explain why.
|
$\left[\begin{array}{rrrr}1 & 2 & 0 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 1 & -3 & 2 \\ 1 & 2 & 1 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrrr}-1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 3 & \frac{1}{2} & -\frac{1}{2} & -\frac{5}{2} \\ -1 & 0 & 0 & 1 \\ -2 & -\frac{3}{4} & \frac{1}{4} & \frac{9}{4}\end{array}\right]$
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{rrrr}1 & 2 & 0 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 1 & -3 & 2 \\ 1 & 2 & 1 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrrr}-1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 3 & \frac{1}{2} & -\frac{1}{2} & -\frac{5}{2} \\ -1 & 0 & 0 & 1 \\ -2 & -\frac{3}{4} & \frac{1}{4} & \frac{9}{4}\end{array}\right]
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.4.12.12
|
A certain river is one half mile wide with a current flowing at 2 miles per hour from East to West. A man swims directly toward the opposite shore from the South bank of the river at a speed of 3 miles per hour. How far down the river does he find himself when he has swam across? How far does he end up traveling?
|
Water: $\left[\begin{array}{ll}-2 & 0\end{array}\right]$ Swimmer: $\left[\begin{array}{ll}0 & 3\end{array}\right]$ Speed relative to earth: $\left[\begin{array}{ll}-2 & 3\end{array}\right]$. It takes him $1 / 6$ of an hour to get across. Therefore, he ends up travelling $\frac{1}{6} \sqrt{4+9}=\frac{1}{6} \sqrt{13}$ miles. He ends up $1 / 3$ mile down stream.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
Water: $\left[\begin{array}{ll}-2 & 0\end{array}\right]$ Swimmer: $\left[\begin{array}{ll}0 & 3\end{array}\right]$ Speed relative to earth: $\left[\begin{array}{ll}-2 & 3\end{array}\right]$. It takes him $1 / 6$ of an hour to get across. Therefore, he ends up travelling $\frac{1}{6} \sqrt{4+9}=\frac{1}{6} \sqrt{13}$ miles. He ends up $1 / 3$ mile down stream.
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.4.12.22
|
An object moves 10 meters in the direction of $\vec{j}$. There are two forces acting on this object, $\vec{F}_{1}=\vec{i}+\vec{j}+2 \vec{k}$, and $\vec{F}_{2}=-5 \vec{i}+2 \vec{j}-6 \vec{k}$. Find the total work done on the object by the two forces. Hint: You can take the work done by the resultant of the two forces or you can add the work done by each force. Why?
|
$\left[\begin{array}{r}-4 \\ 3 \\ -4\end{array}\right] \bullet\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \times 10=30$
You can consider the resultant of the two forces because of the properties of the dot product.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
\left[\begin{array}{r}-4 \\ 3 \\ -4\end{array}\right] \bullet\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \times 10=30$
You can consider the resultant of the two forces because of the properties of the dot product.
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.4.7.9
|
Find the angle between the vectors
$$
\vec{u}=\left[\begin{array}{r}
1 \\
-2 \\
1
\end{array}\right], \vec{v}=\left[\begin{array}{r}
1 \\
2 \\
-7
\end{array}\right]
$$
|
$\frac{-10}{\sqrt{1+4+1} \sqrt{1+4+49}}=-0.55555=\cos \theta$
Therefore we need to solve $-0.55555=\cos \theta$, which gives $\theta=2.0313$ radians.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
\frac{-10}{\sqrt{1+4+1} \sqrt{1+4+49}}=-0.55555=\cos \theta$
Therefore we need to solve $-0.55555=\cos \theta$, which gives $\theta=2.0313$ radians.
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.1.2.13
|
Find $h$ such that
$$
\left[\begin{array}{ll|l}
1 & h & 3 \\
2 & 4 & 6
\end{array}\right]
$$
is the augmented matrix of a consistent system.
|
Any $h$ will work.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
Any $h$ will work.
|
college_math.A_First_Course_in_Linear_Algebra
|
exercise.2.1.15
|
Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{ll}1 & 2 \\ 1 & k\end{array}\right]$. Is it possible to choose $k$ such that $A B=B A$ ? If so, what should $k$ equal?
|
There is no possible choice of $k$ which will make these matrices commute.
|
Creative Commons License (CC BY)
|
college_math.linear_algebra
|
There is no possible choice of $k$ which will make these matrices commute.
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.1.4
|
Find solutions $y_{1}, y_{2}, \ldots, y_{n}$ of the equation $y^{(n)}=0$ that satisfy the initial conditions $y_{i}^{(j)}\left(x_{0}\right)=\left\{\begin{array}{ll}0, & j \neq i-1, \\1, & j=i-1,\end{array} \quad 1 \leq i \leq n .$
|
$y_{i}=\frac{\left(x-x_{0}\right)^{i-1}}{(i-1) !}, 1 \leq i \leq n$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{i}=\frac{\left(x-x_{0}\right)^{i-1}}{(i-1) !}, 1 \leq i \leq n
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.1.17
|
Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $(x-1) y^{\prime \prime}-x y^{\prime}+y=0$, given that $y_{1}=e^{x}$.
|
$y_{2}=x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{2}=x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.24
|
Find a curve $y=y(x)$ through $(2,1)$ such that the normal to the curve at any point $\left(x_{0}, y\left(x_{0}\right)\right)$ intersects the $y$ axis at $y_{I}=2 y\left(x_{0}\right)$.
|
$y=\sqrt{x^{2}-3}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\sqrt{x^{2}-3}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.30
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-4 & 0 & -1 \\ -1 & -3 & -1 \\ 1 & 0 & -2\end{array}\right] \mathbf{y}$
|
\mathbf{y}=\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] e^{-2 t}+c_{2}\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{r}
-1 \\
0 \\
0
\end{array}\right] e^{-2 t}+\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] t e^{-2 t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] e^{-2 t}+c_{2}\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{r}
-1 \\
0 \\
0
\end{array}\right] e^{-2 t}+\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] t e^{-2 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.2.26
|
Solve the equation $y^{\prime}=\frac{\cos x}{\sin y}, \quad y(\pi)=\frac{\pi}{2}$ explicitly. HINT: Use the identity $\cos (x+\pi / 2)=-\sin x$ and the periodicity of the cosine.
|
$y=-x+3 \pi / 2$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-x+3 \pi / 2
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.28
|
Find the orthogonal trajectories of the given family of curves: $x y e^{x^{2}}=c$
|
$y^{2}=-\frac{1}{2} \ln \left(1+2 x^{2}\right)+k$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y^{2}=-\frac{1}{2} \ln \left(1+2 x^{2}\right)+k
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.6
|
A 3200-lb car is moving at $64 \mathrm{ft} / \mathrm{s}$ down a 30-degree grade when it runs out of fuel. Find its velocity after that if friction exerts a resistive force with magnitude proportional to the square of the speed, with $k=1 \mathrm{lb}-\mathrm{s}^{2} / \mathrm{ft}^{2}$. Also find its terminal velocity.
|
$v=-\frac{40\left(13+3 e^{-4 t / 5}\right)}{13-3 e^{-4 t / 5}} ;-40 \mathrm{ft} / \mathrm{s}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
v=-\frac{40\left(13+3 e^{-4 t / 5}\right)}{13-3 e^{-4 t / 5}} ;-40 \mathrm{ft} / \mathrm{s}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.10
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-1 & 1 & -1 \\ -2 & 0 & 2 \\ -1 & 3 & -1\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \frac{e^{-2 t}}{2}+\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] t e^{-2 t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \frac{e^{-2 t}}{2}+\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right] t e^{-2 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.11
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 2 \\ -5 & 1\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}3 \sin 3 t-\cos 3 t \\ 5 \cos 3 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-3 \cos 3 t-\sin 3 t \\ 5 \sin 3 t\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}3 \sin 3 t-\cos 3 t \\ 5 \cos 3 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-3 \cos 3 t-\sin 3 t \\ 5 \sin 3 t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.16
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 2 & -2 \\ 0 & 2 & -1 \\ 1 & 0 & 0\end{array}\right] \mathbf{y}^{\prime}$
|
$\mathbf{y}=c_{1}\left[\begin{array}{r}6 \\ -3 \\ 3\end{array}\right] e^{8 t}+\left[\begin{array}{c}10 \cos 4 t-4 \sin 4 t \\ 17 \cos 4 t-\sin 4 t \\ 3 \cos 4 t-7 \sin 4 t\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{r}6 \\ -3 \\ 3\end{array}\right] e^{8 t}+\left[\begin{array}{c}10 \cos 4 t-4 \sin 4 t \\ 17 \cos 4 t-\sin 4 t \\ 3 \cos 4 t-7 \sin 4 t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.19
|
Two objects suspended from identical springs are set into motion. The period of one object is twice the period of the other. How are the weights of the two objects related?
|
The object with the longer period weighs four times as much as the other.
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
The object with the longer period weighs four times as much as the other.
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.10
|
Find the general solution of the given Euler equation on $(0, \infty)$: $3 x^{2} y^{\prime \prime}-x y^{\prime}+y=0$
|
$y=c_{1} x+c_{2} x^{1 / 3}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} x+c_{2} x^{1 / 3}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.3.30
|
Find the general solution of $y^{\prime \prime}+\omega_{0}^{2} y=M \cos \omega x+N \sin \omega x$, where $M$ and $N$ are constants and $\omega$ and $\omega_{0}$ are distinct positive numbers.
|
$y=\frac{1}{\omega_{0}^{2}-\omega^{2}}(M \cos \omega x+N \sin \omega x)+c_{1} \cos \omega_{0} x+c_{2} \sin \omega_{0} x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{1}{\omega_{0}^{2}-\omega^{2}}(M \cos \omega x+N \sin \omega x)+c_{1} \cos \omega_{0} x+c_{2} \sin \omega_{0} x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.1.1
|
Find the general solution: $y^{\prime}+a y=0(a=$ constant $)$
|
$y=e^{-a x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-a x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.11
|
Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}-3 x y^{\prime}+2 y=0$
|
$y=c_{1} x^{2}+c_{2} x^{1 / 2}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} x^{2}+c_{2} x^{1 / 2}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.1.19
|
Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $4 x^{2}(\sin x) y^{\prime \prime}-4 x(x \cos x+\sin x) y^{\prime}+(2 x \cos x+3 \sin x) y=0$, given that $y_{1}=x^{1 / 2}$.
|
$y_{2}=x^{1 / 2} \cos x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{2}=x^{1 / 2} \cos x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.13
|
A $2 \mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. An external force $F(t)=\sin 8 t \mathrm{lb}$ is applied to the weight, which is released from rest 2 inches below equilibrium. Find its displacement for $t>0$.
|
$y=-t \cos 8 t-\frac{1}{6} \cos 8 t+\frac{1}{8} \sin 8 t \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-t \cos 8 t-\frac{1}{6} \cos 8 t+\frac{1}{8} \sin 8 t \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.3
|
At $12: 00 \mathrm{PM}$ a thermometer reading $10^{\circ} \mathrm{F}$ is placed in a room where the temperature is $70^{\circ} \mathrm{F}$. It reads $56^{\circ}$ when it's placed outside, where the temperature is $5^{\circ} \mathrm{F}$, at $12: 03$. What does it read at 12:05 PM?
|
$\approx 24.33^{\circ} \mathrm{F}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\approx 24.33^{\circ} \mathrm{F}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.3
|
A spring with natural length $.5 \mathrm{~m}$ has length $50.5 \mathrm{~cm}$ with a mass of $2 \mathrm{gm}$ suspended from it. The mass is initially displaced $1.5 \mathrm{~cm}$ below equilibrium and released with zero velocity. Find its displacement for $t>0$.
|
$y=1.5 \cos 14 \sqrt{10} t \mathrm{~cm}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=1.5 \cos 14 \sqrt{10} t \mathrm{~cm}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.6
|
An object is placed in a room where the temperature is $20^{\circ} \mathrm{C}$. The temperature of the object drops by $5^{\circ} \mathrm{C}$ in 4 minutes and by $7^{\circ} \mathrm{C}$ in 8 minutes. What was the temperature of the object when it was initially placed in the room?
|
$(85 / 3)^{\circ} \mathrm{C}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
(85 / 3)^{\circ} \mathrm{C}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.27
|
Find the orthogonal trajectories of the given family of curves: $y=c e^{2 x}$
|
$y^{2}=-x+k$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y^{2}=-x+k
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.31
|
Find a fundamental set of solutions: $\left(D^{2}+9\right)^{3} D^{2} y=0$
|
$\left\{\cos 3 x, x \cos 3 x, x^{2} \cos 3 x, \sin 3 x, x \sin 3 x, x^{2} \sin 3 x, 1, x\right\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\left\{\cos 3 x, x \cos 3 x, x^{2} \cos 3 x, \sin 3 x, x \sin 3 x, x^{2} \sin 3 x, 1, x\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.5
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}4 & 12 \\ -3 & -8\end{array}\right] \mathbf{y}$
|
c_{1}\left[\begin{array}{r}
-2 \\
1
\end{array}\right]+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-2 t}}{3}+\left[\begin{array}{r}
-2 \\
1
\end{array}\right] t e^{-2 t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
c_{1}\left[\begin{array}{r}
-2 \\
1
\end{array}\right]+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-2 t}}{3}+\left[\begin{array}{r}
-2 \\
1
\end{array}\right] t e^{-2 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.2
|
Find the general solution: $x^{2} y^{\prime \prime}+x y^{\prime}-y=\frac{4}{x^{2}} ; \quad y_{1}=x$
|
$y=\frac{4}{3 x^{2}}+c_{1} x+\frac{c_{2}}{x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{4}{3 x^{2}}+c_{1} x+\frac{c_{2}}{x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.32
|
Find a fundamental set of solutions: $(D-2)^{3}(D+1)^{2} D y=0$
|
$\left\{e^{2 x}, x e^{2 x}, x^{2} e^{2 x}, e^{-x}, x e^{-x}, 1\right\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\left\{e^{2 x}, x e^{2 x}, x^{2} e^{2 x}, e^{-x}, x e^{-x}, 1\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.29
|
Find the orthogonal trajectories of the given family of curves: $y=\frac{c e^{x}}{x}$
|
$y^{2}=-2 x-\ln (x-1)^{2}+k$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y^{2}=-2 x-\ln (x-1)^{2}+k
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.6
|
An $8 \mathrm{lb}$ weight stretches a spring $.32 \mathrm{ft}$. The weight is initially displaced 6 inches above equilibrium and given an upward velocity of $4 \mathrm{ft} / \mathrm{sec}$. Find its displacement for $t>0$ if the medium exerts a damping force of $1.5 \mathrm{lb}$ for each $\mathrm{ft} / \mathrm{sec}$ of velocity.
|
$y=\frac{1}{2} e^{-3 t}\left(\cos \sqrt{91} t+\frac{11}{\sqrt{91}} \sin \sqrt{91} t\right) \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{1}{2} e^{-3 t}\left(\cos \sqrt{91} t+\frac{11}{\sqrt{91}} \sin \sqrt{91} t\right) \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.1.15
|
Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $\left(1+3 x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+4 y$.
|
$b_{n}=(n+2)(n+1) a_{n+2}+\left(3 n^{2}-5 n+4\right) a_{n}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
b_{n}=(n+2)(n+1) a_{n+2}+\left(3 n^{2}-5 n+4\right) a_{n}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.20
|
A mass of one $\mathrm{kg}$ stretches a spring $49 \mathrm{~cm}$ in equilibrium. It is attached to a dashpot that supplies a damping force of $4 \mathrm{~N}$ for each $\mathrm{m} / \mathrm{sec}$ of speed. Find the steady state component of its displacement if it's subjected to an external force $F(t)=8 \sin 2 t-6 \cos 2 t \mathrm{~N}$.
|
$y=-\frac{1}{2} \cos 2 t+\frac{1}{4} \sin 2 t \mathrm{~m}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-\frac{1}{2} \cos 2 t+\frac{1}{4} \sin 2 t \mathrm{~m}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.7
|
Find the steady state current in the circuit described by the equation.
$\frac{1}{20} Q^{\prime \prime}+2 Q^{\prime}+100 Q=10 \cos 25 t-5 \sin 25 t$
|
$I_{p}=\frac{20}{37}(\cos 25 t-6 \sin 25 t)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I_{p}=\frac{20}{37}(\cos 25 t-6 \sin 25 t)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.1
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 4 \\ -1 & 7\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
2 \\
1
\end{array}\right] e^{5 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] e^{5 t}+\left[\begin{array}{l}
2 \\
1
\end{array}\right] t e^{5 t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
2 \\
1
\end{array}\right] e^{5 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] e^{5 t}+\left[\begin{array}{l}
2 \\
1
\end{array}\right] t e^{5 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.14
|
Find the general solution: $2 x y^{\prime \prime}+(4 x+1) y^{\prime}+(2 x+1) y=3 x^{1 / 2} e^{-x} ; \quad y_{1}=e^{-x}$
|
$y=e^{-x}\left(x^{3 / 2}+c_{1}+c_{2} x^{1 / 2}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-x}\left(x^{3 / 2}+c_{1}+c_{2} x^{1 / 2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.19
|
Find a fundamental set of solutions: $x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0 ; \quad y_{1}=x^{2}$
|
$\{x^{2}, x^{3}\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\{x^{2}, x^{3}\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.14
|
Find the general solution: $y^{(4)}-4 y^{\prime \prime \prime}+7 y^{\prime \prime}-6 y^{\prime}+2 y=0$
|
$y=e^{x}\left(c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{x}\left(c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.3.60
|
Find the general solution: $y^{\prime \prime \prime}-y^{\prime \prime}-y^{\prime}+y=e^{2 x}(10+3 x)$
|
$y=e^{2 x}(1+x)+c_{1} e^{-x}+e^{x}\left(c_{2}+c_{3} x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{2 x}(1+x)+c_{1} e^{-x}+e^{x}\left(c_{2}+c_{3} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.20
|
Two objects suspended from identical springs are set into motion. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?
|
$T_{2}=\sqrt{2} T_{1}$, where $T_{1}$ is the period of the smaller object.
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
T_{2}=\sqrt{2} T_{1}$, where $T_{1}$ is the period of the smaller object.
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.3
|
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=2$ ohms; $L=.1$ henrys; $C=.01$ farads; $Q_{0}=2$ coulombs; $I_{0}=0$ amperes.
|
$I=-\frac{200}{3} e^{-10 t} \sin 30 t$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I=-\frac{200}{3} e^{-10 t} \sin 30 t
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.11
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}4 & -2 & -2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-2 \\
-3 \\
1
\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] e^{4 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{4 t}}{2}+\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] t e^{4 t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-2 \\
-3 \\
1
\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] e^{4 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{4 t}}{2}+\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] t e^{4 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.16
|
Solve the equation explicitly: $y^{\prime}=\frac{y^{2}+2 x y}{x^{2}}$
|
$y=\frac{c x^{2}}{1-c x} \quad y=-x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{c x^{2}}{1-c x} \quad y=-x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.2
|
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=2$ ohms; $L=.05$ henrys; $C=.01$ farads'; $Q_{0}=2$ coulombs; $I_{0}=-2$ amperes.
|
$I=e^{-20 t}(2 \cos 40 t-101 \sin 40 t)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I=e^{-20 t}(2 \cos 40 t-101 \sin 40 t)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.3
|
A boat weighs $64,000 \mathrm{lb}$. Its propellor produces a constant thrust of $50,000 \mathrm{lb}$ and the water exerts a resistive force with magnitude proportional to the speed, with $k=2000 \mathrm{lb}-\mathrm{s} / \mathrm{ft}$. Assuming that the boat starts from rest, find its velocity as a function of time, and find its terminal velocity.
|
$v=25\left(1-e^{-t}\right) ; 25 \mathrm{ft} / \mathrm{s}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
v=25\left(1-e^{-t}\right) ; 25 \mathrm{ft} / \mathrm{s}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.13
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 1 & 2 \\ 1 & 0 & -1 \\ -1 & -2 & -1\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1}\left[\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right] e^{-2 t}+c_{2} e^{t}\left[\begin{array}{r}\sin t \\ -\cos t \\ \cos t\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}-\cos t \\ -\sin t \\ \sin t\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right] e^{-2 t}+c_{2} e^{t}\left[\begin{array}{r}\sin t \\ -\cos t \\ \cos t\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}-\cos t \\ -\sin t \\ \sin t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.18
|
Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}+10 x y^{\prime}+9 y=0$
|
$y=\frac{1}{x^{2}}\left[c_{1} \cos \left(\frac{1}{\sqrt{2}} \ln x\right)+c_{2} \sin \left(\frac{1}{\sqrt{2}} \ln x\right)\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{1}{x^{2}}\left[c_{1} \cos \left(\frac{1}{\sqrt{2}} \ln x\right)+c_{2} \sin \left(\frac{1}{\sqrt{2}} \ln x\right)\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.1
|
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}+6 x y^{\prime}+6 y=0$
|
$y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}(2 m+1) x^{2 m}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}(m+1) x^{2 m+1}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}(2 m+1) x^{2 m}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}(m+1) x^{2 m+1}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.8
|
A mass of $20 \mathrm{gm}$ stretches a spring $5 \mathrm{~cm}$. The spring is attached to a dashpot with damping constant 400 dyne sec/cm. Determine the displacement for $t>0$ if the mass is initially displaced $9 \mathrm{~cm}$ above equilibrium and released from rest.
|
$y=e^{-10 t}\left(9 \cos 4 \sqrt{6} t+\frac{45}{2 \sqrt{6}} \sin 4 \sqrt{6} t\right) \mathrm{cm}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-10 t}\left(9 \cos 4 \sqrt{6} t+\frac{45}{2 \sqrt{6}} \sin 4 \sqrt{6} t\right) \mathrm{cm}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.9
|
Find the general solution: $x^{2} y^{\prime \prime}+x y^{\prime}-4 y=-6 x-4 ; \quad y_{1}=x^{2}$
|
$y=2 x+1+c_{1} x^{2}+\frac{c_{2}}{x^{2}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=2 x+1+c_{1} x^{2}+\frac{c_{2}}{x^{2}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.21
|
A mass $m$ is suspended from a spring with constant $k$ and subjected to an external force $F(t)=$ $\alpha \cos \omega_{0} t+\beta \sin \omega_{0} t$, where $\omega_{0}$ is the natural frequency of the spring-mass system without damping. Find the steady state component of the displacement if a dashpot with constant $c$ supplies damping.
|
$y_{p}=\frac{1}{c \omega_{0}}\left(-\beta \cos \omega_{0} t+\alpha \sin \omega_{0} t\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{p}=\frac{1}{c \omega_{0}}\left(-\beta \cos \omega_{0} t+\alpha \sin \omega_{0} t\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.23
|
Find a curve $y=y(x)$ through $(0,2)$ such that the normal to the curve at any point $\left(x_{0}, y\left(x_{0}\right)\right)$ intersects the $x$ axis at $x_{I}=x_{0}+1$.
|
$y=\sqrt{2 x+4}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\sqrt{2 x+4}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.1.6
|
Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $y^{\prime \prime}+3\left(x^{2}+1\right) y^{\prime}-2 y=0$, given that $W(\pi)=0$.
|
0
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
0
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.30
|
Find a fundamental set of solutions: $(D-1)^{2}(2 D-1)^{3}\left(D^{2}+1\right) y=0$
|
$\left\{e^{x}, x e^{x}, e^{x / 2}, x e^{x / 2}, x^{2} e^{x / 2}, \cos x, \sin x\right\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\left\{e^{x}, x e^{x}, e^{x / 2}, x e^{x / 2}, x^{2} e^{x / 2}, \cos x, \sin x\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.6
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}-10 & 9 \\ -4 & 2\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
3 \\
2
\end{array}\right] e^{-4 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-4 t}}{2}+\left[\begin{array}{l}
3 \\
2
\end{array}\right] t e^{-4 t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
3 \\
2
\end{array}\right] e^{-4 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-4 t}}{2}+\left[\begin{array}{l}
3 \\
2
\end{array}\right] t e^{-4 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.4
|
Find the general solution: $y^{\prime \prime}-4 y^{\prime}+4 y=0$
|
$y=e^{2 x}\left(c_{1}+c_{2} x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{2 x}\left(c_{1}+c_{2} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.1.7
|
Find the Wronskian $W$ of a set of three solutions of $y^{\prime \prime \prime}+2 x y^{\prime \prime}+e^{x} y^{\prime}-y=0$, given that $W(0)=2$.
|
$2 e^{-x^{2}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
2 e^{-x^{2}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.14
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-x y^{\prime}+10 y=0$
|
$y=x\left[c_{1} \cos (3 \ln x)+c_{2} \sin (3 \ln x)\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x\left[c_{1} \cos (3 \ln x)+c_{2} \sin (3 \ln x)\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.9
|
Find the general solution: $y^{(4)}-16 y=0$
|
$y=c_{1} e^{2 x}+c_{2} e^{-2 x}+c_{3} \cos 2 x+c_{4} \sin 2 x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} e^{2 x}+c_{2} e^{-2 x}+c_{3} \cos 2 x+c_{4} \sin 2 x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.2.24
|
Solve the equation $y^{\prime}=\frac{\left(1+y^{2}\right)}{\left(1+x^{2}\right)}$ explicitly. Hint: Use the identity $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$.
|
$y=\frac{x+c}{1-c x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{x+c}{1-c x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.2
|
Solve the given Bernoulli equation: $7 x y^{\prime}-2 y=-\frac{x^{2}}{y^{6}}$
|
$y=x^{2 / 7}(c-\ln |x|)^{1 / 7}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{2 / 7}(c-\ln |x|)^{1 / 7}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.8
|
Find the general solution: $y^{(4)}+y^{\prime \prime}=0$
|
$y=c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.5
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1}\left[\begin{array}{c}
-1 \\
-1 \\
2
\end{array}\right] e^{-2 t}+c_{2} e^{4 t}\left[\begin{array}{c}
\cos 2 t-\sin 2 t \\
\cos 2 t+\sin 2 t \\
2 \cos 2 t
\end{array}\right]+c_{3} e^{4 t}\left[\begin{array}{c}
\sin 2 t+\cos 2 t \\
\sin 2 t-\cos 2 t \\
2 \sin 2 t
\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{c}
-1 \\
-1 \\
2
\end{array}\right] e^{-2 t}+c_{2} e^{4 t}\left[\begin{array}{c}
\cos 2 t-\sin 2 t \\
\cos 2 t+\sin 2 t \\
2 \cos 2 t
\end{array}\right]+c_{3} e^{4 t}\left[\begin{array}{c}
\sin 2 t+\cos 2 t \\
\sin 2 t-\cos 2 t \\
2 \sin 2 t
\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.12
|
Find the general solution: $10 y^{\prime \prime}-3 y^{\prime}-y=0$
|
$y=c_{1} e^{-x / 5}+c_{2} e^{x / 2}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} e^{-x / 5}+c_{2} e^{x / 2}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.52
|
Find two linearly independent Frobenius solutions of the equation: $4 x^{2} y^{\prime \prime}+2 x\left(4-x^{2}\right) y^{\prime}+\left(1+7 x^{2}\right) y=0$
|
$y_{1}=x^{-1 / 2}\left(1-\frac{1}{2} x^{2}+\frac{1}{32} x^{4}\right)$
$y_{2}=y_{1} \ln x+x^{3 / 2}\left(\frac{5}{8}-\frac{9}{128} x^{2}+\sum_{m=2}^{\infty} \frac{1}{4^{m+1}(m-1) m(m+1)(m+1) !} x^{2 m}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{-1 / 2}\left(1-\frac{1}{2} x^{2}+\frac{1}{32} x^{4}\right)$
$y_{2}=y_{1} \ln x+x^{3 / 2}\left(\frac{5}{8}-\frac{9}{128} x^{2}+\sum_{m=2}^{\infty} \frac{1}{4^{m+1}(m-1) m(m+1)(m+1) !} x^{2 m}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.2
|
A fluid initially at $100^{\circ} \mathrm{C}$ is placed outside on a day when the temperature is $-10^{\circ} \mathrm{C}$, and the temperature of the fluid drops $20^{\circ} \mathrm{C}$ in one minute. Find the temperature $T(t)$ of the fluid for $t>0$.
|
$T=-10+110 e^{-t \ln \frac{11}{9}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
T=-10+110 e^{-t \ln \frac{11}{9}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.4.16
|
Find the general solution for the equation: $y^{\prime \prime}-6 y^{\prime}+8 y=e^{x}(11-6 x)$
|
$y=e^{x}(1-2 x)+c_{1} e^{2 x}+c_{2} e^{4 x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{x}(1-2 x)+c_{1} e^{2 x}+c_{2} e^{4 x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.29
|
Find a fundamental set of solutions: $\left(x^{2}-2 x\right) y^{\prime \prime}+\left(2-x^{2}\right) y^{\prime}+(2 x-2) y=0 ; \quad y_{1}=e^{x}$
|
$\left\{e^{x}, x^{2}\right\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\left\{e^{x}, x^{2}\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.17
|
A mass of one $\mathrm{kg}$ is attached to a spring with constant $k=4 \mathrm{~N} / \mathrm{m}$. An external force $F(t)=$ $-\cos \omega t-2 \sin \omega t \mathrm{n}$ is applied to the mass. Find the displacement $y$ for $t>0$ if $\omega$ equals the natural frequency of the spring-mass system. Assume that the mass is initially displaced $3 \mathrm{~m}$ above equilibrium and given an upward velocity of $450 \mathrm{~cm} / \mathrm{s}$.
|
$y=\frac{t}{2} \cos 2 t-\frac{t}{4} \sin 2 t+3 \cos 2 t+2 \sin 2 t \mathrm{~m}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{t}{2} \cos 2 t-\frac{t}{4} \sin 2 t+3 \cos 2 t+2 \sin 2 t \mathrm{~m}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.3
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}1 & 2 \\ -4 & 5\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c}
\cos 2 t+\sin 2 t \\
2 \cos 2 t
\end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c}
\sin 2 t-\cos 2 t \\
2 \sin 2 t
\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c}
\cos 2 t+\sin 2 t \\
2 \cos 2 t
\end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c}
\sin 2 t-\cos 2 t \\
2 \sin 2 t
\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.1
|
Find the general solution: $y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0$
|
$y=e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.5.8
|
Find a fundamental set of Frobenius solutions for the equation: $18 x^{2}(1+x) y^{\prime \prime}+3 x\left(5+11 x+x^{2}\right) y^{\prime}-\left(1-2 x-5 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution.
|
$y_{1}=x^{1 / 3}\left(1-\frac{1}{3} x+\frac{2}{15} x^{2}-\frac{5}{63} x^{3}+\cdots\right)$
$y_{2}=x^{-1 / 6}\left(1-\frac{1}{12} x^{2}+\frac{1}{18} x^{3}+\cdots\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{1 / 3}\left(1-\frac{1}{3} x+\frac{2}{15} x^{2}-\frac{5}{63} x^{3}+\cdots\right)$
$y_{2}=x^{-1 / 6}\left(1-\frac{1}{12} x^{2}+\frac{1}{18} x^{3}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.7
|
Find the general solution: $y^{\prime \prime}-2 y^{\prime}+2 y=e^{x} \sec x ; \quad y_{1}=e^{x} \cos x$
|
$y=e^{x}\left(x \sin x+\cos x \ln |\cos x|+c_{1} \cos x+c_{2} \sin x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{x}\left(x \sin x+\cos x \ln |\cos x|+c_{1} \cos x+c_{2} \sin x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.2
|
An object stretches a string 1.2 inches in equilibrium. Find its displacement for $t>0$ if it's initially displaced 3 inches below equilibrium and given a downward velocity of $2 \mathrm{ft} / \mathrm{s}$.
|
$y=-\frac{1}{4} \cos 8 \sqrt{5} t-\frac{1}{4 \sqrt{5}} \sin 8 \sqrt{5} t \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-\frac{1}{4} \cos 8 \sqrt{5} t-\frac{1}{4 \sqrt{5}} \sin 8 \sqrt{5} t \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.2
|
Find the general solution: $y^{\prime \prime}-4 y^{\prime}+5 y=0$
|
$y=e^{2 x}\left(c_{1} \cos x+c_{2} \sin x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{2 x}\left(c_{1} \cos x+c_{2} \sin x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.3
|
Find the general solution: $y^{\prime \prime \prime}-y^{\prime \prime}+16 y^{\prime}-16 y=0$
|
$y=c_{1} e^{x}+c_{2} \cos 4 x+c_{3} \sin 4 x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} e^{x}+c_{2} \cos 4 x+c_{3} \sin 4 x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.3.64
|
Find the general solution: $y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=e^{x}(1+x)$
|
$y=\frac{x^{3} e^{x}}{24}(4+x)+e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{x^{3} e^{x}}{24}(4+x)+e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.5
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}+x y^{\prime}+y=0$
|
$y=c_{1} \cos (\ln x)+c_{2} \sin (\ln x)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} \cos (\ln x)+c_{2} \sin (\ln x)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.1
|
A $64 \mathrm{lb}$ object stretches a spring $4 \mathrm{ft}$ in equilibrium. It is attached to a dashpot with damping constant $c=8 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}$. The object is initially displaced 18 inches above equilibrium and given a downward velocity of $4 \mathrm{ft} / \mathrm{sec}$. Find its displacement and time-varying amplitude for $t>0$.
|
$y=\frac{e^{-2 t}}{2}(3 \cos 2 t-\sin 2 t) \mathrm{ft} ; \sqrt{\frac{5}{2}} e^{-2 t} \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{e^{-2 t}}{2}(3 \cos 2 t-\sin 2 t) \mathrm{ft} ; \sqrt{\frac{5}{2}} e^{-2 t} \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.9
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}5 & -4 \\ 10 & 1\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c}\cos 6 t-3 \sin 6 t \\ 5 \cos 6 t\end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c}\sin 6 t+3 \cos 6 t \\ 5 \sin 6 t\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c}\cos 6 t-3 \sin 6 t \\ 5 \cos 6 t\end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c}\sin 6 t+3 \cos 6 t \\ 5 \sin 6 t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.32
|
Solve the given homogeneous equation implicitly: $y^{\prime}=\frac{y}{y-2 x}$
|
$y^{2}(y-3 x)=c ; \quad y \equiv 0 ; y=3 x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y^{2}(y-3 x)=c ; \quad y \equiv 0 ; y=3 x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.1
|
Find the general solution: $y^{\prime \prime}+5 y^{\prime}-6 y=0$
|
$y=c_{1} e^{-6 x}+c_{2} e^{x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} e^{-6 x}+c_{2} e^{x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.1.5
|
Find the general solution: $x^{2} y^{\prime}+y=0$
|
$y=c e^{1 / x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c e^{1 / x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.9
|
Find the general solution of the given Euler equation on $(0, \infty)$: $4 x^{2} y^{\prime \prime}+8 x y^{\prime}+y=0$
|
$y=x^{-1 / 2}\left(c_{1}+c_{2} \ln x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{-1 / 2}\left(c_{1}+c_{2} \ln x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.4
|
Find the power series in $x$ for the general solution: $\left(1-x^{2}\right) y^{\prime \prime}-8 x y^{\prime}-12 y=0$
|
$y=a_{0} \sum_{m=0}^{\infty}(m+1)(2 m+1) x^{2 m}+\frac{a_{1}}{3} \sum_{m=0}^{\infty}(m+1)(2 m+3) x^{2 m+1}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=a_{0} \sum_{m=0}^{\infty}(m+1)(2 m+1) x^{2 m}+\frac{a_{1}}{3} \sum_{m=0}^{\infty}(m+1)(2 m+3) x^{2 m+1}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.1.11
|
Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $y^{\prime \prime}-6 y^{\prime}+9 y=0$, given that $y_{1}=e^{3 x}$.
|
$y_{2}=x e^{3 x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{2}=x e^{3 x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.7
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}-13 & 16 \\ -9 & 11\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
4 \\
3
\end{array}\right] e^{-t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-t}}{3}+\left[\begin{array}{l}
4 \\
3
\end{array}\right] t e^{-t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
4 \\
3
\end{array}\right] e^{-t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] \frac{e^{-t}}{3}+\left[\begin{array}{l}
4 \\
3
\end{array}\right] t e^{-t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.13
|
Find the general solution: $4 y^{(4)}+12 y^{\prime \prime \prime}+3 y^{\prime \prime}-13 y^{\prime}-6 y=0$
|
$y=c_{1} e^{x}+c_{2} e^{-2 x}+c_{3} e^{-x / 2}+c_{4} e^{-3 x / 2}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} e^{x}+c_{2} e^{-2 x}+c_{3} e^{-x / 2}+c_{4} e^{-3 x / 2}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.9
|
Find the general solution: $\mathbf{y}^{\prime}=\frac{1}{3}\left[\begin{array}{rrr}1 & 1 & -3 \\ -4 & -4 & 3 \\ -2 & 1 & 0\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right] e^{t}+c_{2}\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] e^{-t}+c_{3}\left(\left[\begin{array}{l}
0 \\
3 \\
0
\end{array}\right] e^{-t}+\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] t e^{-t}\right)
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right] e^{t}+c_{2}\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] e^{-t}+c_{3}\left(\left[\begin{array}{l}
0 \\
3 \\
0
\end{array}\right] e^{-t}+\left[\begin{array}{r}
1 \\
-1 \\
1
\end{array}\right] t e^{-t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.3.66
|
Find the general solution: $y^{\prime \prime \prime}+2 y^{\prime \prime}-y^{\prime}-2 y=e^{-2 x}[(23-2 x) \cos x+(8-9 x) \sin x]$
|
$y=e^{-2 x}\left[\left(1+\frac{x}{2}\right) \cos x+\left(\frac{3}{2}-2 x\right) \sin x\right]+c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-2 x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-2 x}\left[\left(1+\frac{x}{2}\right) \cos x+\left(\frac{3}{2}-2 x\right) \sin x\right]+c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-2 x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.5.2
|
Find a fundamental set of Frobenius solutions for the equation: $3 x^{2} y^{\prime \prime}+2 x\left(1+x-2 x^{2}\right) y^{\prime}+\left(2 x-8 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution.
|
$y_{1}=x^{1 / 3}\left(1-\frac{2}{3} x+\frac{8}{9} x^{2}-\frac{40}{81} x^{3}+\cdots\right)$
$y_{2}=1-x+\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\cdots$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{1 / 3}\left(1-\frac{2}{3} x+\frac{8}{9} x^{2}-\frac{40}{81} x^{3}+\cdots\right)$
$y_{2}=1-x+\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\cdots
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.5.1
|
Find a fundamental set of Frobenius solutions for the equation: $2 x^{2}\left(1+x+x^{2}\right) y^{\prime \prime}+x\left(3+3 x+5 x^{2}\right) y^{\prime}-y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution.
|
$y_{1}=x^{1 / 2}\left(1-\frac{1}{5} x-\frac{2}{35} x^{2}+\frac{31}{315} x^{3}+\cdots\right)$
$y_{2}=x^{-1}\left(1+x+\frac{1}{2} x^{2}-\frac{1}{6} x^{3}+\cdots\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{1 / 2}\left(1-\frac{1}{5} x-\frac{2}{35} x^{2}+\frac{31}{315} x^{3}+\cdots\right)$
$y_{2}=x^{-1}\left(1+x+\frac{1}{2} x^{2}-\frac{1}{6} x^{3}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.8
|
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}-10 x y^{\prime}+28 y=0$
|
$y=a_{0}\left(1-14 x^{2}+\frac{35}{3} x^{4}\right)+a_{1}\left(x-3 x^{3}+\frac{3}{5} x^{5}+\frac{1}{35} x^{7}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=a_{0}\left(1-14 x^{2}+\frac{35}{3} x^{4}\right)+a_{1}\left(x-3 x^{3}+\frac{3}{5} x^{5}+\frac{1}{35} x^{7}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.3
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-x y^{\prime}+y=0$
|
$y=x\left(c_{1}+c_{2} \ln x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x\left(c_{1}+c_{2} \ln x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.21
|
Find a fundamental set of solutions: $4 x y^{\prime \prime}+2 y^{\prime}+y=0 ; \quad y_{1}=\sin \sqrt{x}$
|
$\{\sin \sqrt{x}, \cos \sqrt{x}\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\{\sin \sqrt{x}, \cos \sqrt{x}\}
|
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