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college_math.A_First_Course_in_Linear_Algebra
exercise.7.2.1
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.
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college_math.linear_algebra
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.
$\boxed{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
Yes. It has a unique solution.
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college_math.linear_algebra
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.
$\boxed{Yes. It has a unique solution.}$
college_math.A_First_Course_in_Linear_Algebra
exercise.2.1.13
$X^{T} Y=\left[\begin{array}{rrr}0 & -1 & -2 \\ 0 & -1 & -2 \\ 0 & 1 & 2\end{array}\right], X Y^{T}=1$
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college_math.linear_algebra
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.
$\boxed{X^{T} Y=\left[\begin{array}{rrr}0 & -1 & -2 \\ 0 & -1 & -2 \\ 0 & 1 & 2\end{array}\right], X Y^{T}=1}$
college_math.A_First_Course_in_Linear_Algebra
exercise.2.1.42
$\left[\begin{array}{rrrr}1 & 2 & 0 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 1 & -3 & 2 \\ 1 & 2 & 1 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrrr}-1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 3 & \frac{1}{2} & -\frac{1}{2} & -\frac{5}{2} \\ -1 & 0 & 0 & 1 \\ -2 & -\frac{3}{4} & \frac{1}{4} & \frac{9}{4}\end{array}\right]$
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college_math.linear_algebra
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.
$\boxed{\left[\begin{array}{rrrr}1 & 2 & 0 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 1 & -3 & 2 \\ 1 & 2 & 1 & 2\end{array}\right]^{-1}=\left[\begin{array}{rrrr}-1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 3 & \frac{1}{2} & -\frac{1}{2} & -\frac{5}{2} \\ -1 & 0 & 0 & 1 \\ -2 & -\frac{3}{4} & \frac{1}{4} & \frac{9}{4}\end{array}\right]}$
college_math.A_First_Course_in_Linear_Algebra
exercise.4.12.12
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.
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college_math.linear_algebra
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?
$\boxed{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
$\left[\begin{array}{r}-4 \\ 3 \\ -4\end{array}\right] \bullet\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \times 10=30$ You can consider the resultant of the two forces because of the properties of the dot product.
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college_math.linear_algebra
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?
$\boxed{\left[\begin{array}{r}-4 \\ 3 \\ -4\end{array}\right] \bullet\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \times 10=30$ 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
$\frac{-10}{\sqrt{1+4+1} \sqrt{1+4+49}}=-0.55555=\cos \theta$ Therefore we need to solve $-0.55555=\cos \theta$, which gives $\theta=2.0313$ radians.
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college_math.linear_algebra
Find the angle between the vectors $$ \vec{u}=\left[\begin{array}{r} 1 \\ -2 \\ 1 \end{array}\right], \vec{v}=\left[\begin{array}{r} 1 \\ 2 \\ -7 \end{array}\right] $$
$\boxed{\frac{-10}{\sqrt{1+4+1} \sqrt{1+4+49}}=-0.55555=\cos \theta$ 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
Any $h$ will work.
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college_math.linear_algebra
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.
$\boxed{Any $h$ will work.}$
college_math.A_First_Course_in_Linear_Algebra
exercise.2.1.15
There is no possible choice of $k$ which will make these matrices commute.
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college_math.linear_algebra
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?
$\boxed{There is no possible choice of $k$ which will make these matrices commute.}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.1.4
$y_{i}=\frac{\left(x-x_{0}\right)^{i-1}}{(i-1) !}, 1 \leq i \leq n$
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college_math.differential_equation
Find solutions $y_{1}, y_{2}, \ldots, y_{n}$ of the equation $y^{(n)}=0$ that satisfy the initial conditions $y_{i}^{(j)}\left(x_{0}\right)=\left\{\begin{array}{ll}0, & j \neq i-1, \\1, & j=i-1,\end{array} \quad 1 \leq i \leq n .$
$\boxed{y_{i}=\frac{\left(x-x_{0}\right)^{i-1}}{(i-1) !}, 1 \leq i \leq n}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.1.17
$y_{2}=x$
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college_math.differential_equation
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 $y_{1}=e^{x}$.
$\boxed{y_{2}=x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.5.24
$y=\sqrt{x^{2}-3}$
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college_math.differential_equation
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 $y_{I}=2 y\left(x_{0}\right)$.
$\boxed{y=\sqrt{x^{2}-3}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.30
\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)
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-4 & 0 & -1 \\ -1 & -3 & -1 \\ 1 & 0 & -2\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=\left[\begin{array}{r} -1 \\ 1 \\ 0 \end{array}\right] e^{-2 t}+c_{2}\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{r} -1 \\ 0 \\ 0 \end{array}\right] e^{-2 t}+\left[\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right] t e^{-2 t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.2.26
$y=-x+3 \pi / 2$
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college_math.differential_equation
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.
$\boxed{y=-x+3 \pi / 2}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.5.28
$y^{2}=-\frac{1}{2} \ln \left(1+2 x^{2}\right)+k$
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college_math.differential_equation
Find the orthogonal trajectories of the given family of curves: $x y e^{x^{2}}=c$
$\boxed{y^{2}=-\frac{1}{2} \ln \left(1+2 x^{2}\right)+k}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.3.6
$v=-\frac{40\left(13+3 e^{-4 t / 5}\right)}{13-3 e^{-4 t / 5}} ;-40 \mathrm{ft} / \mathrm{s}$
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college_math.differential_equation
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.
$\boxed{v=-\frac{40\left(13+3 e^{-4 t / 5}\right)}{13-3 e^{-4 t / 5}} ;-40 \mathrm{ft} / \mathrm{s}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.10
\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
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-1 & 1 & -1 \\ -2 & 0 & 2 \\ -1 & 3 & -1\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right] e^{-2 t}+c_{3}\left(\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right] \frac{e^{-2 t}}{2}+\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right] t e^{-2 t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.6.11
$\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}3 \sin 3 t-\cos 3 t \\ 5 \cos 3 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-3 \cos 3 t-\sin 3 t \\ 5 \sin 3 t\end{array}\right]$
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 2 \\ -5 & 1\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}3 \sin 3 t-\cos 3 t \\ 5 \cos 3 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-3 \cos 3 t-\sin 3 t \\ 5 \sin 3 t\end{array}\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.6.16
$\mathbf{y}=c_{1}\left[\begin{array}{r}6 \\ -3 \\ 3\end{array}\right] e^{8 t}+\left[\begin{array}{c}10 \cos 4 t-4 \sin 4 t \\ 17 \cos 4 t-\sin 4 t \\ 3 \cos 4 t-7 \sin 4 t\end{array}\right]$
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 2 & -2 \\ 0 & 2 & -1 \\ 1 & 0 & 0\end{array}\right] \mathbf{y}^{\prime}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{r}6 \\ -3 \\ 3\end{array}\right] e^{8 t}+\left[\begin{array}{c}10 \cos 4 t-4 \sin 4 t \\ 17 \cos 4 t-\sin 4 t \\ 3 \cos 4 t-7 \sin 4 t\end{array}\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.1.19
The object with the longer period weighs four times as much as the other.
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college_math.differential_equation
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?
$\boxed{The object with the longer period weighs four times as much as the other.}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.10
$y=c_{1} x+c_{2} x^{1 / 3}$
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college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $3 x^{2} y^{\prime \prime}-x y^{\prime}+y=0$
$\boxed{y=c_{1} x+c_{2} x^{1 / 3}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.3.30
$y=\frac{1}{\omega_{0}^{2}-\omega^{2}}(M \cos \omega x+N \sin \omega x)+c_{1} \cos \omega_{0} x+c_{2} \sin \omega_{0} x$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
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.
$\boxed{y=\frac{1}{\omega_{0}^{2}-\omega^{2}}(M \cos \omega x+N \sin \omega x)+c_{1} \cos \omega_{0} x+c_{2} \sin \omega_{0} x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.1.1
$y=e^{-a x}$
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college_math.differential_equation
Find the general solution: $y^{\prime}+a y=0(a=$ constant $)$
$\boxed{y=e^{-a x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.11
$y=c_{1} x^{2}+c_{2} x^{1 / 2}$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}-3 x y^{\prime}+2 y=0$
$\boxed{y=c_{1} x^{2}+c_{2} x^{1 / 2}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.1.19
$y_{2}=x^{1 / 2} \cos x$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
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 $y_{1}=x^{1 / 2}$.
$\boxed{y_{2}=x^{1 / 2} \cos x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.1.13
$y=-t \cos 8 t-\frac{1}{6} \cos 8 t+\frac{1}{8} \sin 8 t \mathrm{ft}$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
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 $t>0$.
$\boxed{y=-t \cos 8 t-\frac{1}{6} \cos 8 t+\frac{1}{8} \sin 8 t \mathrm{ft}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.2.3
$\approx 24.33^{\circ} \mathrm{F}$
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college_math.differential_equation
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?
$\boxed{\approx 24.33^{\circ} \mathrm{F}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.1.3
$y=1.5 \cos 14 \sqrt{10} t \mathrm{~cm}$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
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 $t>0$.
$\boxed{y=1.5 \cos 14 \sqrt{10} t \mathrm{~cm}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.2.6
$(85 / 3)^{\circ} \mathrm{C}$
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college_math.differential_equation
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?
$\boxed{(85 / 3)^{\circ} \mathrm{C}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.5.27
$y^{2}=-x+k$
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college_math.differential_equation
Find the orthogonal trajectories of the given family of curves: $y=c e^{2 x}$
$\boxed{y^{2}=-x+k}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.31
$\left\{\cos 3 x, x \cos 3 x, x^{2} \cos 3 x, \sin 3 x, x \sin 3 x, x^{2} \sin 3 x, 1, x\right\}$
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college_math.differential_equation
Find a fundamental set of solutions: $\left(D^{2}+9\right)^{3} D^{2} y=0$
$\boxed{\left\{\cos 3 x, x \cos 3 x, x^{2} \cos 3 x, \sin 3 x, x \sin 3 x, x^{2} \sin 3 x, 1, x\right\}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.5
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
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}4 & 12 \\ -3 & -8\end{array}\right] \mathbf{y}$
$\boxed{c_{1}\left[\begin{array}{r} -2 \\ 1 \end{array}\right]+c_{2}\left(\left[\begin{array}{r} -1 \\ 0 \end{array}\right] \frac{e^{-2 t}}{3}+\left[\begin{array}{r} -2 \\ 1 \end{array}\right] t e^{-2 t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.2
$y=\frac{4}{3 x^{2}}+c_{1} x+\frac{c_{2}}{x}$
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college_math.differential_equation
Find the general solution: $x^{2} y^{\prime \prime}+x y^{\prime}-y=\frac{4}{x^{2}} ; \quad y_{1}=x$
$\boxed{y=\frac{4}{3 x^{2}}+c_{1} x+\frac{c_{2}}{x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.32
$\left\{e^{2 x}, x e^{2 x}, x^{2} e^{2 x}, e^{-x}, x e^{-x}, 1\right\}$
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college_math.differential_equation
Find a fundamental set of solutions: $(D-2)^{3}(D+1)^{2} D y=0$
$\boxed{\left\{e^{2 x}, x e^{2 x}, x^{2} e^{2 x}, e^{-x}, x e^{-x}, 1\right\}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.5.29
$y^{2}=-2 x-\ln (x-1)^{2}+k$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
Find the orthogonal trajectories of the given family of curves: $y=\frac{c e^{x}}{x}$
$\boxed{y^{2}=-2 x-\ln (x-1)^{2}+k}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.2.6
$y=\frac{1}{2} e^{-3 t}\left(\cos \sqrt{91} t+\frac{11}{\sqrt{91}} \sin \sqrt{91} t\right) \mathrm{ft}$
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college_math.differential_equation
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.
$\boxed{y=\frac{1}{2} e^{-3 t}\left(\cos \sqrt{91} t+\frac{11}{\sqrt{91}} \sin \sqrt{91} t\right) \mathrm{ft}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.1.15
$b_{n}=(n+2)(n+1) a_{n+2}+\left(3 n^{2}-5 n+4\right) a_{n}$
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college_math.differential_equation
Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $\left(1+3 x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+4 y$.
$\boxed{b_{n}=(n+2)(n+1) a_{n+2}+\left(3 n^{2}-5 n+4\right) a_{n}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.2.20
$y=-\frac{1}{2} \cos 2 t+\frac{1}{4} \sin 2 t \mathrm{~m}$
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college_math.differential_equation
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 $F(t)=8 \sin 2 t-6 \cos 2 t \mathrm{~N}$.
$\boxed{y=-\frac{1}{2} \cos 2 t+\frac{1}{4} \sin 2 t \mathrm{~m}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.3.7
$I_{p}=\frac{20}{37}(\cos 25 t-6 \sin 25 t)$
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college_math.differential_equation
Find the steady state current in the circuit described by the equation. $\frac{1}{20} Q^{\prime \prime}+2 Q^{\prime}+100 Q=10 \cos 25 t-5 \sin 25 t$
$\boxed{I_{p}=\frac{20}{37}(\cos 25 t-6 \sin 25 t)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.1
\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)
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 4 \\ -1 & 7\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{l} 2 \\ 1 \end{array}\right] e^{5 t}+c_{2}\left(\left[\begin{array}{r} -1 \\ 0 \end{array}\right] e^{5 t}+\left[\begin{array}{l} 2 \\ 1 \end{array}\right] t e^{5 t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.14
$y=e^{-x}\left(x^{3 / 2}+c_{1}+c_{2} x^{1 / 2}\right)$
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college_math.differential_equation
Find the general solution: $2 x y^{\prime \prime}+(4 x+1) y^{\prime}+(2 x+1) y=3 x^{1 / 2} e^{-x} ; \quad y_{1}=e^{-x}$
$\boxed{y=e^{-x}\left(x^{3 / 2}+c_{1}+c_{2} x^{1 / 2}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.19
$\{x^{2}, x^{3}\}$
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college_math.differential_equation
Find a fundamental set of solutions: $x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0 ; \quad y_{1}=x^{2}$
$\boxed{\{x^{2}, x^{3}\}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.14
$y=e^{x}\left(c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x\right)$
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college_math.differential_equation
Find the general solution: $y^{(4)}-4 y^{\prime \prime \prime}+7 y^{\prime \prime}-6 y^{\prime}+2 y=0$
$\boxed{y=e^{x}\left(c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.3.60
$y=e^{2 x}(1+x)+c_{1} e^{-x}+e^{x}\left(c_{2}+c_{3} x\right)$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime \prime}-y^{\prime \prime}-y^{\prime}+y=e^{2 x}(10+3 x)$
$\boxed{y=e^{2 x}(1+x)+c_{1} e^{-x}+e^{x}\left(c_{2}+c_{3} x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.1.20
$T_{2}=\sqrt{2} T_{1}$, where $T_{1}$ is the period of the smaller object.
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college_math.differential_equation
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?
$\boxed{T_{2}=\sqrt{2} T_{1}$, where $T_{1}$ is the period of the smaller object.}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.3.3
$I=-\frac{200}{3} e^{-10 t} \sin 30 t$
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college_math.differential_equation
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.
$\boxed{I=-\frac{200}{3} e^{-10 t} \sin 30 t}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.11
\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)
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}4 & -2 & -2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{r} -2 \\ -3 \\ 1 \end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right] e^{4 t}+c_{3}\left(\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] \frac{e^{4 t}}{2}+\left[\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right] t e^{4 t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.4.16
$y=\frac{c x^{2}}{1-c x} \quad y=-x$
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college_math.differential_equation
Solve the equation explicitly: $y^{\prime}=\frac{y^{2}+2 x y}{x^{2}}$
$\boxed{y=\frac{c x^{2}}{1-c x} \quad y=-x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.3.2
$I=e^{-20 t}(2 \cos 40 t-101 \sin 40 t)$
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college_math.differential_equation
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.
$\boxed{I=e^{-20 t}(2 \cos 40 t-101 \sin 40 t)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.3.3
$v=25\left(1-e^{-t}\right) ; 25 \mathrm{ft} / \mathrm{s}$
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college_math.differential_equation
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.
$\boxed{v=25\left(1-e^{-t}\right) ; 25 \mathrm{ft} / \mathrm{s}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.6.13
$\mathbf{y}=c_{1}\left[\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right] e^{-2 t}+c_{2} e^{t}\left[\begin{array}{r}\sin t \\ -\cos t \\ \cos t\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}-\cos t \\ -\sin t \\ \sin t\end{array}\right]$
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 1 & 2 \\ 1 & 0 & -1 \\ -1 & -2 & -1\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right] e^{-2 t}+c_{2} e^{t}\left[\begin{array}{r}\sin t \\ -\cos t \\ \cos t\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}-\cos t \\ -\sin t \\ \sin t\end{array}\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.18
$y=\frac{1}{x^{2}}\left[c_{1} \cos \left(\frac{1}{\sqrt{2}} \ln x\right)+c_{2} \sin \left(\frac{1}{\sqrt{2}} \ln x\right)\right]$
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college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}+10 x y^{\prime}+9 y=0$
$\boxed{y=\frac{1}{x^{2}}\left[c_{1} \cos \left(\frac{1}{\sqrt{2}} \ln x\right)+c_{2} \sin \left(\frac{1}{\sqrt{2}} \ln x\right)\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.2.1
$y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}(2 m+1) x^{2 m}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}(m+1) x^{2 m+1}$
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college_math.differential_equation
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}+6 x y^{\prime}+6 y=0$
$\boxed{y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}(2 m+1) x^{2 m}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}(m+1) x^{2 m+1}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.2.8
$y=e^{-10 t}\left(9 \cos 4 \sqrt{6} t+\frac{45}{2 \sqrt{6}} \sin 4 \sqrt{6} t\right) \mathrm{cm}$
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college_math.differential_equation
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.
$\boxed{y=e^{-10 t}\left(9 \cos 4 \sqrt{6} t+\frac{45}{2 \sqrt{6}} \sin 4 \sqrt{6} t\right) \mathrm{cm}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.9
$y=2 x+1+c_{1} x^{2}+\frac{c_{2}}{x^{2}}$
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college_math.differential_equation
Find the general solution: $x^{2} y^{\prime \prime}+x y^{\prime}-4 y=-6 x-4 ; \quad y_{1}=x^{2}$
$\boxed{y=2 x+1+c_{1} x^{2}+\frac{c_{2}}{x^{2}}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.2.21
$y_{p}=\frac{1}{c \omega_{0}}\left(-\beta \cos \omega_{0} t+\alpha \sin \omega_{0} t\right)$
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college_math.differential_equation
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.
$\boxed{y_{p}=\frac{1}{c \omega_{0}}\left(-\beta \cos \omega_{0} t+\alpha \sin \omega_{0} t\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.5.23
$y=\sqrt{2 x+4}$
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college_math.differential_equation
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 $x_{I}=x_{0}+1$.
$\boxed{y=\sqrt{2 x+4}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.1.6
0
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college_math.differential_equation
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 $W(\pi)=0$.
$\boxed{0}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.30
$\left\{e^{x}, x e^{x}, e^{x / 2}, x e^{x / 2}, x^{2} e^{x / 2}, \cos x, \sin x\right\}$
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college_math.differential_equation
Find a fundamental set of solutions: $(D-1)^{2}(2 D-1)^{3}\left(D^{2}+1\right) y=0$
$\boxed{\left\{e^{x}, x e^{x}, e^{x / 2}, x e^{x / 2}, x^{2} e^{x / 2}, \cos x, \sin x\right\}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.6
\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)
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}-10 & 9 \\ -4 & 2\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{l} 3 \\ 2 \end{array}\right] e^{-4 t}+c_{2}\left(\left[\begin{array}{r} -1 \\ 0 \end{array}\right] \frac{e^{-4 t}}{2}+\left[\begin{array}{l} 3 \\ 2 \end{array}\right] t e^{-4 t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.2.4
$y=e^{2 x}\left(c_{1}+c_{2} x\right)$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime}-4 y^{\prime}+4 y=0$
$\boxed{y=e^{2 x}\left(c_{1}+c_{2} x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.1.7
$2 e^{-x^{2}}$
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college_math.differential_equation
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 $W(0)=2$.
$\boxed{2 e^{-x^{2}}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.14
$y=x\left[c_{1} \cos (3 \ln x)+c_{2} \sin (3 \ln x)\right]$
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college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-x y^{\prime}+10 y=0$
$\boxed{y=x\left[c_{1} \cos (3 \ln x)+c_{2} \sin (3 \ln x)\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.9
$y=c_{1} e^{2 x}+c_{2} e^{-2 x}+c_{3} \cos 2 x+c_{4} \sin 2 x$
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college_math.differential_equation
Find the general solution: $y^{(4)}-16 y=0$
$\boxed{y=c_{1} e^{2 x}+c_{2} e^{-2 x}+c_{3} \cos 2 x+c_{4} \sin 2 x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.2.24
$y=\frac{x+c}{1-c x}$
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college_math.differential_equation
Solve the equation $y^{\prime}=\frac{\left(1+y^{2}\right)}{\left(1+x^{2}\right)}$ explicitly. Hint: Use the identity $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$.
$\boxed{y=\frac{x+c}{1-c x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.4.2
$y=x^{2 / 7}(c-\ln |x|)^{1 / 7}$
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college_math.differential_equation
Solve the given Bernoulli equation: $7 x y^{\prime}-2 y=-\frac{x^{2}}{y^{6}}$
$\boxed{y=x^{2 / 7}(c-\ln |x|)^{1 / 7}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.8
$y=c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x$
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college_math.differential_equation
Find the general solution: $y^{(4)}+y^{\prime \prime}=0$
$\boxed{y=c_{1}+c_{2} x+c_{3} \cos x+c_{4} \sin x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.6.5
$\mathbf{y}=c_{1}\left[\begin{array}{c} -1 \\ -1 \\ 2 \end{array}\right] e^{-2 t}+c_{2} e^{4 t}\left[\begin{array}{c} \cos 2 t-\sin 2 t \\ \cos 2 t+\sin 2 t \\ 2 \cos 2 t \end{array}\right]+c_{3} e^{4 t}\left[\begin{array}{c} \sin 2 t+\cos 2 t \\ \sin 2 t-\cos 2 t \\ 2 \sin 2 t \end{array}\right]$
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{c} -1 \\ -1 \\ 2 \end{array}\right] e^{-2 t}+c_{2} e^{4 t}\left[\begin{array}{c} \cos 2 t-\sin 2 t \\ \cos 2 t+\sin 2 t \\ 2 \cos 2 t \end{array}\right]+c_{3} e^{4 t}\left[\begin{array}{c} \sin 2 t+\cos 2 t \\ \sin 2 t-\cos 2 t \\ 2 \sin 2 t \end{array}\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.2.12
$y=c_{1} e^{-x / 5}+c_{2} e^{x / 2}$
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college_math.differential_equation
Find the general solution: $10 y^{\prime \prime}-3 y^{\prime}-y=0$
$\boxed{y=c_{1} e^{-x / 5}+c_{2} e^{x / 2}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.6.52
$y_{1}=x^{-1 / 2}\left(1-\frac{1}{2} x^{2}+\frac{1}{32} x^{4}\right)$ $y_{2}=y_{1} \ln x+x^{3 / 2}\left(\frac{5}{8}-\frac{9}{128} x^{2}+\sum_{m=2}^{\infty} \frac{1}{4^{m+1}(m-1) m(m+1)(m+1) !} x^{2 m}\right)$
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
college_math.differential_equation
Find two linearly independent Frobenius solutions of the equation: $4 x^{2} y^{\prime \prime}+2 x\left(4-x^{2}\right) y^{\prime}+\left(1+7 x^{2}\right) y=0$
$\boxed{y_{1}=x^{-1 / 2}\left(1-\frac{1}{2} x^{2}+\frac{1}{32} x^{4}\right)$ $y_{2}=y_{1} \ln x+x^{3 / 2}\left(\frac{5}{8}-\frac{9}{128} x^{2}+\sum_{m=2}^{\infty} \frac{1}{4^{m+1}(m-1) m(m+1)(m+1) !} x^{2 m}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.4.2.2
$T=-10+110 e^{-t \ln \frac{11}{9}}$
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college_math.differential_equation
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 $t>0$.
$\boxed{T=-10+110 e^{-t \ln \frac{11}{9}}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.4.16
$y=e^{x}(1-2 x)+c_{1} e^{2 x}+c_{2} e^{4 x}$
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college_math.differential_equation
Find the general solution for the equation: $y^{\prime \prime}-6 y^{\prime}+8 y=e^{x}(11-6 x)$
$\boxed{y=e^{x}(1-2 x)+c_{1} e^{2 x}+c_{2} e^{4 x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.29
$\left\{e^{x}, x^{2}\right\}$
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college_math.differential_equation
Find a fundamental set of solutions: $\left(x^{2}-2 x\right) y^{\prime \prime}+\left(2-x^{2}\right) y^{\prime}+(2 x-2) y=0 ; \quad y_{1}=e^{x}$
$\boxed{\left\{e^{x}, x^{2}\right\}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.1.17
$y=\frac{t}{2} \cos 2 t-\frac{t}{4} \sin 2 t+3 \cos 2 t+2 \sin 2 t \mathrm{~m}$
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college_math.differential_equation
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 $450 \mathrm{~cm} / \mathrm{s}$.
$\boxed{y=\frac{t}{2} \cos 2 t-\frac{t}{4} \sin 2 t+3 \cos 2 t+2 \sin 2 t \mathrm{~m}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.6.3
$\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c} \cos 2 t+\sin 2 t \\ 2 \cos 2 t \end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c} \sin 2 t-\cos 2 t \\ 2 \sin 2 t \end{array}\right]$
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}1 & 2 \\ -4 & 5\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c} \cos 2 t+\sin 2 t \\ 2 \cos 2 t \end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c} \sin 2 t-\cos 2 t \\ 2 \sin 2 t \end{array}\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.1
$y=e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0$
$\boxed{y=e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.5.8
$y_{1}=x^{1 / 3}\left(1-\frac{1}{3} x+\frac{2}{15} x^{2}-\frac{5}{63} x^{3}+\cdots\right)$ $y_{2}=x^{-1 / 6}\left(1-\frac{1}{12} x^{2}+\frac{1}{18} x^{3}+\cdots\right)$
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college_math.differential_equation
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.
$\boxed{y_{1}=x^{1 / 3}\left(1-\frac{1}{3} x+\frac{2}{15} x^{2}-\frac{5}{63} x^{3}+\cdots\right)$ $y_{2}=x^{-1 / 6}\left(1-\frac{1}{12} x^{2}+\frac{1}{18} x^{3}+\cdots\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.7
$y=e^{x}\left(x \sin x+\cos x \ln |\cos x|+c_{1} \cos x+c_{2} \sin x\right)$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime}-2 y^{\prime}+2 y=e^{x} \sec x ; \quad y_{1}=e^{x} \cos x$
$\boxed{y=e^{x}\left(x \sin x+\cos x \ln |\cos x|+c_{1} \cos x+c_{2} \sin x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.1.2
$y=-\frac{1}{4} \cos 8 \sqrt{5} t-\frac{1}{4 \sqrt{5}} \sin 8 \sqrt{5} t \mathrm{ft}$
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college_math.differential_equation
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 $2 \mathrm{ft} / \mathrm{s}$.
$\boxed{y=-\frac{1}{4} \cos 8 \sqrt{5} t-\frac{1}{4 \sqrt{5}} \sin 8 \sqrt{5} t \mathrm{ft}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.2.2
$y=e^{2 x}\left(c_{1} \cos x+c_{2} \sin x\right)$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime}-4 y^{\prime}+5 y=0$
$\boxed{y=e^{2 x}\left(c_{1} \cos x+c_{2} \sin x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.3
$y=c_{1} e^{x}+c_{2} \cos 4 x+c_{3} \sin 4 x$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime \prime}-y^{\prime \prime}+16 y^{\prime}-16 y=0$
$\boxed{y=c_{1} e^{x}+c_{2} \cos 4 x+c_{3} \sin 4 x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.3.64
$y=\frac{x^{3} e^{x}}{24}(4+x)+e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=e^{x}(1+x)$
$\boxed{y=\frac{x^{3} e^{x}}{24}(4+x)+e^{x}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.5
$y=c_{1} \cos (\ln x)+c_{2} \sin (\ln x)$
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college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}+x y^{\prime}+y=0$
$\boxed{y=c_{1} \cos (\ln x)+c_{2} \sin (\ln x)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.6.2.1
$y=\frac{e^{-2 t}}{2}(3 \cos 2 t-\sin 2 t) \mathrm{ft} ; \sqrt{\frac{5}{2}} e^{-2 t} \mathrm{ft}$
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college_math.differential_equation
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 $t>0$.
$\boxed{y=\frac{e^{-2 t}}{2}(3 \cos 2 t-\sin 2 t) \mathrm{ft} ; \sqrt{\frac{5}{2}} e^{-2 t} \mathrm{ft}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.6.9
$\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c}\cos 6 t-3 \sin 6 t \\ 5 \cos 6 t\end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c}\sin 6 t+3 \cos 6 t \\ 5 \sin 6 t\end{array}\right]$
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}5 & -4 \\ 10 & 1\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1} e^{3 t}\left[\begin{array}{c}\cos 6 t-3 \sin 6 t \\ 5 \cos 6 t\end{array}\right]+c_{2} e^{3 t}\left[\begin{array}{c}\sin 6 t+3 \cos 6 t \\ 5 \sin 6 t\end{array}\right]}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.4.32
$y^{2}(y-3 x)=c ; \quad y \equiv 0 ; y=3 x$
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college_math.differential_equation
Solve the given homogeneous equation implicitly: $y^{\prime}=\frac{y}{y-2 x}$
$\boxed{y^{2}(y-3 x)=c ; \quad y \equiv 0 ; y=3 x}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.2.1
$y=c_{1} e^{-6 x}+c_{2} e^{x}$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime}+5 y^{\prime}-6 y=0$
$\boxed{y=c_{1} e^{-6 x}+c_{2} e^{x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.2.1.5
$y=c e^{1 / x}$
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college_math.differential_equation
Find the general solution: $x^{2} y^{\prime}+y=0$
$\boxed{y=c e^{1 / x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.9
$y=x^{-1 / 2}\left(c_{1}+c_{2} \ln x\right)$
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college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $4 x^{2} y^{\prime \prime}+8 x y^{\prime}+y=0$
$\boxed{y=x^{-1 / 2}\left(c_{1}+c_{2} \ln x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.2.4
$y=a_{0} \sum_{m=0}^{\infty}(m+1)(2 m+1) x^{2 m}+\frac{a_{1}}{3} \sum_{m=0}^{\infty}(m+1)(2 m+3) x^{2 m+1}$
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college_math.differential_equation
Find the power series in $x$ for the general solution: $\left(1-x^{2}\right) y^{\prime \prime}-8 x y^{\prime}-12 y=0$
$\boxed{y=a_{0} \sum_{m=0}^{\infty}(m+1)(2 m+1) x^{2 m}+\frac{a_{1}}{3} \sum_{m=0}^{\infty}(m+1)(2 m+3) x^{2 m+1}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.1.11
$y_{2}=x e^{3 x}$
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college_math.differential_equation
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 $y_{1}=e^{3 x}$.
$\boxed{y_{2}=x e^{3 x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.7
\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)
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}-13 & 16 \\ -9 & 11\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{l} 4 \\ 3 \end{array}\right] e^{-t}+c_{2}\left(\left[\begin{array}{r} -1 \\ 0 \end{array}\right] \frac{e^{-t}}{3}+\left[\begin{array}{l} 4 \\ 3 \end{array}\right] t e^{-t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.2.13
$y=c_{1} e^{x}+c_{2} e^{-2 x}+c_{3} e^{-x / 2}+c_{4} e^{-3 x / 2}$
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college_math.differential_equation
Find the general solution: $4 y^{(4)}+12 y^{\prime \prime \prime}+3 y^{\prime \prime}-13 y^{\prime}-6 y=0$
$\boxed{y=c_{1} e^{x}+c_{2} e^{-2 x}+c_{3} e^{-x / 2}+c_{4} e^{-3 x / 2}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.10.5.9
\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)
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college_math.differential_equation
Find the general solution: $\mathbf{y}^{\prime}=\frac{1}{3}\left[\begin{array}{rrr}1 & 1 & -3 \\ -4 & -4 & 3 \\ -2 & 1 & 0\end{array}\right] \mathbf{y}$
$\boxed{\mathbf{y}=c_{1}\left[\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right] e^{t}+c_{2}\left[\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right] e^{-t}+c_{3}\left(\left[\begin{array}{l} 0 \\ 3 \\ 0 \end{array}\right] e^{-t}+\left[\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right] t e^{-t}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.9.3.66
$y=e^{-2 x}\left[\left(1+\frac{x}{2}\right) \cos x+\left(\frac{3}{2}-2 x\right) \sin x\right]+c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-2 x}$
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college_math.differential_equation
Find the general solution: $y^{\prime \prime \prime}+2 y^{\prime \prime}-y^{\prime}-2 y=e^{-2 x}[(23-2 x) \cos x+(8-9 x) \sin x]$
$\boxed{y=e^{-2 x}\left[\left(1+\frac{x}{2}\right) \cos x+\left(\frac{3}{2}-2 x\right) \sin x\right]+c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-2 x}}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.5.2
$y_{1}=x^{1 / 3}\left(1-\frac{2}{3} x+\frac{8}{9} x^{2}-\frac{40}{81} x^{3}+\cdots\right)$ $y_{2}=1-x+\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\cdots$
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college_math.differential_equation
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.
$\boxed{y_{1}=x^{1 / 3}\left(1-\frac{2}{3} x+\frac{8}{9} x^{2}-\frac{40}{81} x^{3}+\cdots\right)$ $y_{2}=1-x+\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\cdots}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.5.1
$y_{1}=x^{1 / 2}\left(1-\frac{1}{5} x-\frac{2}{35} x^{2}+\frac{31}{315} x^{3}+\cdots\right)$ $y_{2}=x^{-1}\left(1+x+\frac{1}{2} x^{2}-\frac{1}{6} x^{3}+\cdots\right)$
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college_math.differential_equation
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.
$\boxed{y_{1}=x^{1 / 2}\left(1-\frac{1}{5} x-\frac{2}{35} x^{2}+\frac{31}{315} x^{3}+\cdots\right)$ $y_{2}=x^{-1}\left(1+x+\frac{1}{2} x^{2}-\frac{1}{6} x^{3}+\cdots\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.2.8
$y=a_{0}\left(1-14 x^{2}+\frac{35}{3} x^{4}\right)+a_{1}\left(x-3 x^{3}+\frac{3}{5} x^{5}+\frac{1}{35} x^{7}\right)$
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college_math.differential_equation
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}-10 x y^{\prime}+28 y=0$
$\boxed{y=a_{0}\left(1-14 x^{2}+\frac{35}{3} x^{4}\right)+a_{1}\left(x-3 x^{3}+\frac{3}{5} x^{5}+\frac{1}{35} x^{7}\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.7.4.3
$y=x\left(c_{1}+c_{2} \ln x\right)$
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college_math.differential_equation
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-x y^{\prime}+y=0$
$\boxed{y=x\left(c_{1}+c_{2} \ln x\right)}$
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
exercise.5.6.21
$\{\sin \sqrt{x}, \cos \sqrt{x}\}$
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college_math.differential_equation
Find a fundamental set of solutions: $4 x y^{\prime \prime}+2 y^{\prime}+y=0 ; \quad y_{1}=\sin \sqrt{x}$
$\boxed{\{\sin \sqrt{x}, \cos \sqrt{x}\}}$