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college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.1
|
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=3$ ohms; $L=.1$ henrys; $C=.01$ farads; $Q_{0}=0$ coulombs; $I_{0}=2$ amperes.
|
$I=e^{-15 t}\left(2 \cos 5 \sqrt{15} t-\frac{6}{\sqrt{31}} \sin 5 \sqrt{31} t\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I=e^{-15 t}\left(2 \cos 5 \sqrt{15} t-\frac{6}{\sqrt{31}} \sin 5 \sqrt{31} t\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.51
|
Find two linearly independent Frobenius solutions of the equation: $x\left(1+x^{2}\right) y^{\prime \prime}+\left(1-x^{2}\right) y^{\prime}-8 x y=0$
|
$y_{1}=\left(1+x^{2}\right)^{2}$
$y_{2}=y_{1} \ln x-\frac{3}{2} x^{2}-\frac{3}{2} x^{4}+\sum_{m=3}^{\infty} \frac{(-1)^{m}}{m(m-1)(m-2)} x^{2 m}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=\left(1+x^{2}\right)^{2}$
$y_{2}=y_{1} \ln x-\frac{3}{2} x^{2}-\frac{3}{2} x^{4}+\sum_{m=3}^{\infty} \frac{(-1)^{m}}{m(m-1)(m-2)} x^{2 m}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.2
|
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}+2 x y^{\prime}-2 y=0$
|
$y=a_{0} \sum_{m=0}^{\infty}(-1)^{m+1} \frac{x^{2 m}}{2 m-1}+a_{1} x$
|
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+1} \frac{x^{2 m}}{2 m-1}+a_{1} x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.7
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=0$
|
$y=c_{1} x+\frac{c_{2}}{x^{3}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} x+\frac{c_{2}}{x^{3}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.10
|
Find the general solution: $\mathbf{y}^{\prime}=\frac{1}{3}\left[\begin{array}{rr}7 & -5 \\ 2 & 5\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}\cos t-3 \sin t \\ 2 \cos t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}\sin t+3 \cos t \\ 2 \sin t\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}\cos t-3 \sin t \\ 2 \cos t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}\sin t+3 \cos t \\ 2 \sin t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.36
|
Find a fundamental set of solutions: $D^{3}(D-2)^{2}\left(D^{2}+4\right)^{2} y=0$
|
$\left\{1, x, x^{2}, e^{2 x}, x e^{2 x}, \cos 2 x, x \cos 2 x, \sin 2 x, x \sin 2 x\right\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\left\{1, x, x^{2}, e^{2 x}, x e^{2 x}, \cos 2 x, x \cos 2 x, \sin 2 x, x \sin 2 x\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.3.68
|
Find the general solution: $y^{\prime \prime \prime}-4 y^{\prime \prime}+14 y^{\prime \prime}-20 y^{\prime}+25 y=e^{x}[(2+6 x) \cos 2 x+3 \sin 2 x]$
|
$y=-\frac{x^{2} e^{x}}{16}(1+x) \cos 2 x+e^{x}\left[\left(c_{1}+c_{2} x\right) \cos 2 x+\left(c_{3}+c_{4} x\right) \sin 2 x\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-\frac{x^{2} e^{x}}{16}(1+x) \cos 2 x+e^{x}\left[\left(c_{1}+c_{2} x\right) \cos 2 x+\left(c_{3}+c_{4} x\right) \sin 2 x\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.33
|
Solve the given homogeneous equation implicitly: $y^{\prime}=\frac{x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}}$
|
$(x-y)^{3}(x+y)=c y^{2} x^{4} ; \quad y=0 ; y=x ; y=-x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
(x-y)^{3}(x+y)=c y^{2} x^{4} ; \quad y=0 ; y=x ; y=-x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.46
|
Find two linearly independent Frobenius solutions of the equation: $x^{2}(1-x) y^{\prime \prime}+x(3-2 x) y^{\prime}+(1+2 x) y=0$
|
$y_{1}=\frac{(x-1)^{2}}{x}$
$y_{2}=y_{1} \ln x+3-3 x+2 \sum_{n=2}^{\infty} \frac{1}{n\left(n^{2}-1\right)} x^{n}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=\frac{(x-1)^{2}}{x}$
$y_{2}=y_{1} \ln x+3-3 x+2 \sum_{n=2}^{\infty} \frac{1}{n\left(n^{2}-1\right)} x^{n}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.1.8
|
Find the Wronskian $W$ of a set of four solutions of $y^{(4)}+(\tan x) y^{\prime \prime \prime}+x^{2} y^{\prime \prime}+2 x y=0$, given that $W(\pi / 4)=K$.
|
$\sqrt{2} K \cos x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\sqrt{2} K \cos x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.3
|
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}-8 x y^{\prime}+20 y=0$
|
$y=a_{0}\left(1-10 x^{2}+5 x^{4}\right)+a_{1}\left(x-2 x^{3}+\frac{1}{5} x^{5}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=a_{0}\left(1-10 x^{2}+5 x^{4}\right)+a_{1}\left(x-2 x^{3}+\frac{1}{5} x^{5}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.2
|
A firefighter who weighs $192 \mathrm{lb}$ slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to her speed, with constant of proportionality $k$. Find $k$, given that her terminal velocity is $-16 \mathrm{ft} / \mathrm{s}$, and then find her velocity $v$ as a function of $t$. Assume that she starts from rest.
|
$k=12 ; \quad v=-16\left(1-e^{-2 t}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
k=12 ; \quad v=-16\left(1-e^{-2 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.5.10
|
Find a fundamental set of Frobenius solutions for the equation: $10 x^{2}\left(1+x+2 x^{2}\right) y^{\prime \prime}+x\left(13+13 x+66 x^{2}\right) y^{\prime}-\left(1+4 x+10 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution.
|
$y_{1}=x^{1 / 2}\left(1+\frac{3}{17} x-\frac{7}{153} x^{2}-\frac{547}{5661} x^{3}+\cdots\right)$
$y_{2}=x^{-1 / 2}\left(1+x+\frac{14}{13} x^{2}-\frac{556}{897} x^{3}+\cdots\right)$
|
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{3}{17} x-\frac{7}{153} x^{2}-\frac{547}{5661} x^{3}+\cdots\right)$
$y_{2}=x^{-1 / 2}\left(1+x+\frac{14}{13} x^{2}-\frac{556}{897} x^{3}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.8
|
Find the steady state current in the circuit described by the equation.
$\frac{1}{10} Q^{\prime \prime}+2 Q^{\prime}+100 Q=3 \cos 50 t-6 \sin 50 t$
|
$I_{p}=\frac{3}{13}(8 \cos 50 t-\sin 50 t)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I_{p}=\frac{3}{13}(8 \cos 50 t-\sin 50 t)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.10
|
An object weighing $256 \mathrm{lb}$ is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The magnitude of the resisting force is $1 \mathrm{lb}$ when $|v|=4 \mathrm{ft} / \mathrm{s}$. Find $v$ for $t>0$, and find its terminal velocity.
|
$v=-\frac{64\left(1-e^{-t}\right)}{1+e^{-t}} ;-64 \mathrm{ft} / \mathrm{s}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
v=-\frac{64\left(1-e^{-t}\right)}{1+e^{-t}} ;-64 \mathrm{ft} / \mathrm{s}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.4
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{ll}5 & -6 \\ 3 & -1\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}
\cos 3 t-\sin 3 t \\
\cos 3 t
\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}
\sin 3 t+\cos 3 t \\
\sin 3 t
\end{array}\right]$
|
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}
\cos 3 t-\sin 3 t \\
\cos 3 t
\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}
\sin 3 t+\cos 3 t \\
\sin 3 t
\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.5
|
A stone weighing $1 / 2 \mathrm{lb}$ is thrown upward from an initial height of $5 \mathrm{ft}$ with an initial speed of 32 $\mathrm{ft} / \mathrm{s}$. Air resistance is proportional to speed, with $k=1 / 128 \mathrm{lb}-\mathrm{s} / \mathrm{ft}$. Find the maximum height attained by the stone.
|
$\approx 17.10 \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\approx 17.10 \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.5.4
|
Find a fundamental set of Frobenius solutions for the equation: $4 x^{2} y^{\prime \prime}+x\left(7+2 x+4 x^{2}\right) y^{\prime}-\left(1-4 x-7 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution.
|
$y_{1}=x^{1 / 4}\left(1-\frac{1}{2} x-\frac{19}{104} x^{2}+\frac{1571}{10608} x^{3}+\cdots\right)$
$y_{2}=x^{-1}\left(1+2 x-\frac{11}{6} x^{2}-\frac{1}{7} x^{3}+\cdots\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{1 / 4}\left(1-\frac{1}{2} x-\frac{19}{104} x^{2}+\frac{1571}{10608} x^{3}+\cdots\right)$
$y_{2}=x^{-1}\left(1+2 x-\frac{11}{6} x^{2}-\frac{1}{7} x^{3}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.20
|
Find all curves $y=y(x)$ such that the tangent to the curve at any point passes through a given point $\left(x_{1}, y_{1}\right)$.
|
$y=y_{1}+c\left(x-x_{1}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=y_{1}+c\left(x-x_{1}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.1
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}+7 x y^{\prime}+8 y=0$
|
$y=c_{1} x^{-4}+c_{2} x^{-2}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} x^{-4}+c_{2} x^{-2}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.7
|
A $32 \mathrm{lb}$ weight stretches a spring $2 \mathrm{ft}$ in equilibrium. It is attached to a dashpot with constant $c=8$ $\mathrm{lb}-\mathrm{sec} / \mathrm{ft}$. The weight is initially displaced 8 inches below equilibrium and released from rest. Find its displacement for $t>0$.
|
$y=-\frac{e^{-4 t}}{3}(2+8 t)$ ft
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-\frac{e^{-4 t}}{3}(2+8 t)$ ft
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.5
|
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=4$ ohms; $L=.05$ henrys; $C=.008$ farads; $Q_{0}=-1$ coulombs; $I_{0}=2$ amperes.
|
$I=-e^{-40 t}(2 \cos 30 t-86 \sin 30 t)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I=-e^{-40 t}(2 \cos 30 t-86 \sin 30 t)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.9
|
A tank initially contains a solution of 10 pounds of salt in 60 gallons of water. Water with $1 / 2$ pound of salt per gallon is added to the tank at $6 \mathrm{gal} / \mathrm{min}$, and the resulting solution leaves at the same rate. Find the quantity $Q(t)$ of salt in the tank at time $t>0$.
|
$Q(t)=30-20 e^{-t / 10}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
Q(t)=30-20 e^{-t / 10}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.20
|
Find a fundamental set of solutions: $x^{2}(\ln |x|)^{2} y^{\prime \prime}-(2 x \ln |x|) y^{\prime}+(2+\ln |x|) y=0 ; \quad y_{1}=\ln |x|$
|
$\{\ln |x|, x \ln |x|\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\{\ln |x|, x \ln |x|\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.16
|
Find the general solution: $4 x^{2} y^{\prime \prime}-4 x(x+1) y^{\prime}+(2 x+3) y=4 x^{5 / 2} e^{2 x} ; \quad y_{1}=x^{1 / 2}$
|
$y=x^{1 / 2}\left(\frac{e^{2 x}}{2}+c_{1}+c_{2} e^{x}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{1 / 2}\left(\frac{e^{2 x}}{2}+c_{1}+c_{2} e^{x}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.12
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}34 & 52 \\ -20 & -30\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}\sin 4 t-8 \cos 4 t \\ 5 \cos 4 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-\cos 4 t-8 \sin 4 t \\ 5 \sin 4 t\end{array}\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}\sin 4 t-8 \cos 4 t \\ 5 \cos 4 t\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}-\cos 4 t-8 \sin 4 t \\ 5 \sin 4 t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.10
|
Find the general solution: $x^{2} y^{\prime \prime}+2 x(x-1) y^{\prime}+\left(x^{2}-2 x+2\right) y=x^{3} e^{2 x} ; \quad y_{1}=x e^{-x}$
|
$y=\frac{x e^{2 x}}{9}+x e^{-x}\left(c_{1}+c_{2} x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{x e^{2 x}}{9}+x e^{-x}\left(c_{1}+c_{2} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.5.34
|
Find conditions on the constants $A, B, C, D, E$, and $F$ such that the equation
$\left(A x^{2}+B x y+C y^{2}\right) d x+\left(D x^{2}+E x y+F y^{2}\right) d y=0$
is exact.
|
$B=2 D, \quad E=2 C$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
B=2 D, \quad E=2 C
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.17
|
Solve the equation explicitly: $x y^{3} y^{\prime}=y^{4}+x^{4}$
|
$y= \pm x(4 \ln |x|+c)^{1 / 4}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y= \pm x(4 \ln |x|+c)^{1 / 4}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.1
|
A firefighter who weighs $192 \mathrm{lb}$ slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to his speed, with $k=2.5 \mathrm{lb}-\mathrm{s} / \mathrm{ft}$. Assuming that he starts from rest, find his velocity as a function of time and find his terminal velocity.
|
$v=-\frac{384}{5}\left(1-e^{-5 t / 12}\right) ;-\frac{384}{5} \mathrm{ft} / \mathrm{s}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
v=-\frac{384}{5}\left(1-e^{-5 t / 12}\right) ;-\frac{384}{5} \mathrm{ft} / \mathrm{s}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.3
|
Find the general solution: $y^{\prime \prime}+8 y^{\prime}+7 y=0$
|
$y=c_{1} e^{-7 x}+c_{2} e^{-x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} e^{-7 x}+c_{2} e^{-x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.2
|
Find the general solution: $y^{(4)}+8 y^{\prime \prime}-9 y=0$
|
$y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} \cos 3 x+c_{4} \sin 3 x$
|
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} e^{-x}+c_{3} \cos 3 x+c_{4} \sin 3 x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.17
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0$
|
$y=x^{2}\left(c_{1}+c_{2} \ln x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{2}\left(c_{1}+c_{2} \ln x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.24
|
Find a fundamental set of solutions: $x^{2} y^{\prime \prime}-2 x y^{\prime}+\left(x^{2}+2\right) y=0 ; \quad y_{1}=x \sin x$
|
$\{x \sin x, x \cos x\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\{x \sin x, x \cos x\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.4
|
An object stretches a spring 6 inches in equilibrium. Find its displacement for $t>0$ if it's initially displaced 3 inches above equilibrium and given a downward velocity of 6 inches/s. Find the frequency, period, amplitude and phase angle of the motion.
|
$y=\frac{1}{4} \cos 8 t-\frac{1}{16} \sin 8 t \mathrm{ft} ; R=\frac{\sqrt{17}}{16} \mathrm{ft} ; \omega_{0}=8 \mathrm{rad} / \mathrm{s} ; T=\pi / 4 \mathrm{~s}$; $\phi \approx-.245 \mathrm{rad} \approx-14.04^{\circ}$
|
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 t-\frac{1}{16} \sin 8 t \mathrm{ft} ; R=\frac{\sqrt{17}}{16} \mathrm{ft} ; \omega_{0}=8 \mathrm{rad} / \mathrm{s} ; T=\pi / 4 \mathrm{~s}$; $\phi \approx-.245 \mathrm{rad} \approx-14.04^{\circ}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.6
|
Find the power series in $x$ for the general solution: $\left(1+x^{2}\right) y^{\prime \prime}+2 x y^{\prime}+\frac{1}{4} y=0$
|
$y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} \frac{(4 j+1)^{2}}{2 j+1}\right] \frac{x^{2 m}}{8^{m} m !}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} \frac{(4 j+3)^{2}}{2 j+3}\right] \frac{x^{2 m+1}}{8^{m} m !}$
|
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}\left[\prod_{j=0}^{m-1} \frac{(4 j+1)^{2}}{2 j+1}\right] \frac{x^{2 m}}{8^{m} m !}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} \frac{(4 j+3)^{2}}{2 j+3}\right] \frac{x^{2 m+1}}{8^{m} m !}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.28
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-2 & -12 & 10 \\ 2 & -24 & 11 \\ 2 & -24 & 8\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-2 \\
1 \\
2
\end{array}\right] e^{-6 t}+c_{2}\left(-\left[\begin{array}{l}
6 \\
1 \\
0
\end{array}\right] \frac{e^{-6 t}}{6}+\left[\begin{array}{r}
-2 \\
1 \\
2
\end{array}\right] t e^{-6 t}\right)+c_{3}\left(-\left[\begin{array}{c}
12 \\
1 \\
0
\end{array}\right] \frac{e^{-6 t}}{36}-\left[\begin{array}{l}
6 \\
1 \\
0
\end{array}\right] \frac{t e^{-6 t}}{6}+\left[\begin{array}{r}
-2 \\
1 \\
2
\end{array}\right] \frac{t^{2} e^{-6 t}}{2}\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 \\
1 \\
2
\end{array}\right] e^{-6 t}+c_{2}\left(-\left[\begin{array}{l}
6 \\
1 \\
0
\end{array}\right] \frac{e^{-6 t}}{6}+\left[\begin{array}{r}
-2 \\
1 \\
2
\end{array}\right] t e^{-6 t}\right)+c_{3}\left(-\left[\begin{array}{c}
12 \\
1 \\
0
\end{array}\right] \frac{e^{-6 t}}{36}-\left[\begin{array}{l}
6 \\
1 \\
0
\end{array}\right] \frac{t e^{-6 t}}{6}+\left[\begin{array}{r}
-2 \\
1 \\
2
\end{array}\right] \frac{t^{2} e^{-6 t}}{2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.5
|
Find the general solution: $y^{\prime \prime}+2 y^{\prime}+10 y=0$
|
$y=e^{-x}\left(c_{1} \cos 3 x+c_{2} \sin 3 x\right)$
|
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} \cos 3 x+c_{2} \sin 3 x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.32
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-3 & -1 & 0 \\ 1 & -1 & 0 \\ -1 & -1 & -2\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
0 \\
1
\end{array}\right] e^{-3 t}+c_{2}\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] e^{-3 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] e^{-3 t}+\left[\begin{array}{l}
-1 \\
-1 \\
1
\end{array}\right] t e^{-3 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 \\
0 \\
1
\end{array}\right] e^{-3 t}+c_{2}\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] e^{-3 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] e^{-3 t}+\left[\begin{array}{l}
-1 \\
-1 \\
1
\end{array}\right] t e^{-3 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.31
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-3 & -3 & 4 \\ 4 & 5 & -8 \\ 2 & 3 & -5\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
2 \\
0 \\
1
\end{array}\right] e^{-t}+c_{2}\left[\begin{array}{r}
-3 \\
2 \\
0
\end{array}\right] e^{-t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{-t}}{2}+\left[\begin{array}{r}
-1 \\
2 \\
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}{l}
2 \\
0 \\
1
\end{array}\right] e^{-t}+c_{2}\left[\begin{array}{r}
-3 \\
2 \\
0
\end{array}\right] e^{-t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{-t}}{2}+\left[\begin{array}{r}
-1 \\
2 \\
1
\end{array}\right] t e^{-t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.4.15
|
Find the general solution for the equation: $y^{\prime \prime}-3 y^{\prime}+2 y=e^{3 x}(1+x)$
|
$y=\frac{e^{3 x}}{4}(-1+2 x)+c_{1} e^{x}+c_{2} e^{2 x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{e^{3 x}}{4}(-1+2 x)+c_{1} e^{x}+c_{2} e^{2 x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.1.16
|
Suppose $y(x)=\sum_{n=0}^{\infty} a_{n}(x+1)^{n}$ on an open interval that contains $x_{0}=-1$. Find a power series in $x+1$ for $x y^{\prime \prime}+(4+2 x) y^{\prime}+(2+x) y$.
|
$b_{0}=-2 a_{2}+2 a_{1}+a_{0}$,
$b_{n}=-(n+2)(n+1) a_{n+2}+(n+1)(n+2) a_{n+1}+(2 n+1) a_{n}+a_{n-1}, n \geq 2$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
b_{0}=-2 a_{2}+2 a_{1}+a_{0}$,
$b_{n}=-(n+2)(n+1) a_{n+2}+(n+1)(n+2) a_{n+1}+(2 n+1) a_{n}+a_{n-1}, n \geq 2
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.18
|
Solve the equation explicitly: $y^{\prime}=\frac{y}{x}+\sec \frac{y}{x}$
|
$y=x \sin ^{-1}(\ln |x|+c)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x \sin ^{-1}(\ln |x|+c)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.10
|
A tank initially contains 100 liters of a salt solution with a concentration of $.1 \mathrm{~g} / \mathrm{liter}$. A solution with a salt concentration of $.3 \mathrm{~g} / \mathrm{liter}$ is added to the tank at 5 liters $/ \mathrm{min}$, and the resulting mixture is drained out at the same rate. Find the concentration $K(t)$ of salt in the tank as a function of $t$.
|
$K(t)=.3-.2 e^{-t / 20}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
K(t)=.3-.2 e^{-t / 20}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.6
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-3 x y^{\prime}+13 y=0$
|
$y=x^{2}\left[c_{1} \cos (3 \ln x)+c_{2} \sin (3 \ln x)\right]$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{2}\left[c_{1} \cos (3 \ln x)+c_{2} \sin (3 \ln x)\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.22
|
Find a fundamental set of solutions: $x y^{\prime \prime}-(2 x+2) y^{\prime}+(x+2) y=0 ; \quad y_{1}=e^{x}$
|
$\{e^{x}, x^{3} e^{x}\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\{e^{x}, x^{3} e^{x}\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.4
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}3 & 1 \\ -1 & 1\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
1
\end{array}\right] e^{2 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] e^{2 t}+\left[\begin{array}{r}
-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}=c_{1}\left[\begin{array}{r}
-1 \\
1
\end{array}\right] e^{2 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] e^{2 t}+\left[\begin{array}{r}
-1 \\
1
\end{array}\right] t e^{2 t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.3.65
|
Find the general solution: $y^{(4)}-2 y^{\prime \prime}+y=-e^{-x}\left(4-9 x+3 x^{2}\right)$
|
$y=\frac{x^{2} e^{-x}}{16}\left(1+2 x-x^{2}\right)+e^{x}\left(c_{1}+c_{2} x\right)+e^{-x}\left(c_{3}+c_{4} x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{x^{2} e^{-x}}{16}\left(1+2 x-x^{2}\right)+e^{x}\left(c_{1}+c_{2} x\right)+e^{-x}\left(c_{3}+c_{4} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.18
|
Control mechanisms allow fluid to flow into a tank at a rate proportional to the volume $V$ of fluid in the tank, and to flow out at a rate proportional to $V^{2}$. Suppose $V(0)=V_{0}$ and the constants of proportionality are $a$ and $b$, respectively. Find $V(t)$ for $t>0$ and find $\lim _{t \rightarrow \infty} V(t)$.
|
$V=\frac{a}{b} \frac{V_{0}}{V_{0}-\left(V_{0}-a / b\right) e^{-a t}}, \quad \lim _{t \rightarrow \infty} V(t)=a / b$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
V=\frac{a}{b} \frac{V_{0}}{V_{0}-\left(V_{0}-a / b\right) e^{-a t}}, \quad \lim _{t \rightarrow \infty} V(t)=a / b
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.17
|
Find the general solution: $x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=4 x^{2} ; \quad y_{1}=x^{2}$
|
$y=-2 x^{2} \ln x+c_{1} x^{2}+c_{2} x^{4}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-2 x^{2} \ln x+c_{1} x^{2}+c_{2} x^{4}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.3.4
|
Find the current in the $R L C$ circuit, assuming that $E(t)=0$ for $t>0$.
$R=6$ ohms; $L=.1$ henrys; $C=.004$ farads'; $Q_{0}=3$ coulombs; $I_{0}=-10$ amperes.
|
$I=-10 e^{-30 t}(\cos 40 t+18 \sin 40 t)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
I=-10 e^{-30 t}(\cos 40 t+18 \sin 40 t)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.1
|
Find the general solution: $(2 x+1) y^{\prime \prime}-2 y^{\prime}-(2 x+3) y=(2 x+1)^{2} ; \quad y_{1}=e^{-x}$
|
$y=1-2 x+c_{1} e^{-x}+c_{2} x e^{x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=1-2 x+c_{1} e^{-x}+c_{2} x e^{x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.4
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0$
|
$y=x^{-2}\left(c_{1}+c_{2} \ln x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{-2}\left(c_{1}+c_{2} \ln x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.5.19
|
Find all curves $y=y(x)$ such that the tangent to the curve at any point $\left(x_{0}, y\left(x_{0}\right)\right)$ intersects the $x$ axis at $x_{I}=x_{0}^{3}$.
|
$y=\frac{c x}{\sqrt{\left|x^{2}-1\right|}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{c x}{\sqrt{\left|x^{2}-1\right|}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.14
|
A $10 \mathrm{gm}$ mass suspended on a spring moves in simple harmonic motion with period $4 \mathrm{~s}$. Find the period of the simple harmonic motion of a 20 gm mass suspended from the same spring.
|
$T=4 \sqrt{2} \mathrm{~s}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
T=4 \sqrt{2} \mathrm{~s}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.2.35
|
Solve the equation using variation of parameters followed by separation of variables: $y^{\prime}+y=\frac{2 x e^{-x}}{1+y e^{x}}$
|
$y=e^{-x}\left(-1 \pm \sqrt{2 x^{2}+c}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-x}\left(-1 \pm \sqrt{2 x^{2}+c}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.47
|
Find two linearly independent Frobenius solutions of the equation: $4 x^{2}(1+x) y^{\prime \prime}-4 x^{2} y^{\prime}+(1-5 x) y=0$
|
$y_{1}=x^{1 / 2}(x+1)^{2}$
$y_{2}=y_{1} \ln x-x^{3 / 2}\left(3+3 x+2 \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n\left(n^{2}-1\right)} x^{n}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{1 / 2}(x+1)^{2}$
$y_{2}=y_{1} \ln x-x^{3 / 2}\left(3+3 x+2 \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n\left(n^{2}-1\right)} x^{n}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.8
|
A tank initially contains 40 gallons of pure water. A solution with 1 gram of salt per gallon of water is added to the tank at $3 \mathrm{gal} / \mathrm{min}$, and the resulting solution drains out at the same rate. Find the quantity $Q(t)$ of salt in the tank at time $t>0$.
|
$Q(t)=40\left(1-e^{-3 t / 40}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
Q(t)=40\left(1-e^{-3 t / 40}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.6
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-3 & 3 & 1 \\ 1 & -5 & -3 \\ -3 & 7 & 3\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1}\left[\begin{array}{c}
-1 \\
-1 \\
1
\end{array}\right] e^{-t}+c_{2} e^{-2 t}\left[\begin{array}{c}
\cos 2 t-\sin 2 t \\
-\cos 2 t-\sin 2 t \\
2 \cos 2 t
\end{array}\right]+c_{3} e^{-2 t}\left[\begin{array}{c}
\sin 2 t+\cos 2 t \\
-\sin 2 t+\cos 2 t \\
2 \sin 2 t
\end{array}\right]$
|
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 \\
1
\end{array}\right] e^{-t}+c_{2} e^{-2 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^{-2 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.5.2.10
|
Find the general solution: $y^{\prime \prime}+6 y^{\prime}+13 y=0$
|
$y=e^{-3 x}\left(c_{1} \cos 2 x+c_{2} \sin 2 x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-3 x}\left(c_{1} \cos 2 x+c_{2} \sin 2 x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.2.4
|
Find all solutions: $x^{2} y y^{\prime}=\left(y^{2}-1\right)^{3 / 2}$
|
$\frac{(\ln y)^{2}}{2}=-\frac{x^{3}}{3}+c$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\frac{(\ln y)^{2}}{2}=-\frac{x^{3}}{3}+c
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.9
|
Find the general solution: $y^{\prime \prime}-2 y^{\prime}+3 y=0$
|
$y=e^{x}\left(c_{1} \cos \sqrt{2} x+c_{2} \sin \sqrt{2} x\right)$
|
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} \cos \sqrt{2} x+c_{2} \sin \sqrt{2} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.11
|
An $8 \mathrm{lb}$ weight stretches a spring 2 inches. It is attached to a dashpot with damping constant $c=4 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}$. The weight is initially displaced 3 inches above equilibrium and given a downward velocity of $4 \mathrm{ft} / \mathrm{sec}$. Find its displacement for $t>0$.
|
$y=e^{-8 t}\left(\frac{1}{4} \cos 8 \sqrt{2} t-\frac{1}{4 \sqrt{2}} \sin 8 \sqrt{2} t\right) \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-8 t}\left(\frac{1}{4} \cos 8 \sqrt{2} t-\frac{1}{4 \sqrt{2}} \sin 8 \sqrt{2} t\right) \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.1.3
|
A mass $m_{1}$ is suspended from a rigid support on a spring $S_{1}$ with spring constant $k_{1}$ and damping constant $c_{1}$. A second mass $m_{2}$ is suspended from the first on a spring $S_{2}$ with spring constant $k_{2}$ and damping constant $c_{2}$, and a third mass $m_{3}$ is suspended from the second on a spring $S_{3}$ with spring constant $k_{3}$ and damping constant $c_{3}$. Let $y_{1}=y_{1}(t), y_{2}=y_{2}(t)$, and $y_{3}=y_{3}(t)$ be the displacements of the three masses from their equilibrium positions at time $t$, measured positive upward. Derive a system of differential equations for $y_{1}, y_{2}$ and $y_{3}$, assuming that the masses of the springs are negligible and that vertical external forces $F_{1}, F_{2}$, and $F_{3}$ also act on the masses.
|
\begin{align*}
m_{1} y_{1}^{\prime \prime}&=-\left(c_{1}+c_{2}\right) y_{1}^{\prime}+c_{2} y_{2}^{\prime}-\left(k_{1}+k_{2}\right) y_{1}+k_{2} y_{2}+F_{1} \\
m_{2} y_{2}^{\prime \prime}&=\left(c_{2}-c_{3}\right) y_{1}^{\prime}-\left(c_{2}+c_{3}\right) y_{2}^{\prime}+c_{3} y_{3}^{\prime}+\left(k_{2}-k_{3}\right) y_{1}-\left(k_{2}+k_{3}\right) y_{2}+k_{3} y_{3}+F_{2} \\
m_{3} y_{3}^{\prime \prime}&=c_{3} y_{1}^{\prime}+c_{3} y_{2}^{\prime}-c_{3} y_{3}^{\prime}+k_{3} y_{1}+k_{3} y_{2}-k_{3} y_{3}+F_{3}
\end{align*}
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\begin{align*}
m_{1} y_{1}^{\prime \prime}&=-\left(c_{1}+c_{2}\right) y_{1}^{\prime}+c_{2} y_{2}^{\prime}-\left(k_{1}+k_{2}\right) y_{1}+k_{2} y_{2}+F_{1} \\
m_{2} y_{2}^{\prime \prime}&=\left(c_{2}-c_{3}\right) y_{1}^{\prime}-\left(c_{2}+c_{3}\right) y_{2}^{\prime}+c_{3} y_{3}^{\prime}+\left(k_{2}-k_{3}\right) y_{1}-\left(k_{2}+k_{3}\right) y_{2}+k_{3} y_{3}+F_{2} \\
m_{3} y_{3}^{\prime \prime}&=c_{3} y_{1}^{\prime}+c_{3} y_{2}^{\prime}-c_{3} y_{3}^{\prime}+k_{3} y_{1}+k_{3} y_{2}-k_{3} y_{3}+F_{3}
\end{align*}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.5.7
|
Find a fundamental set of Frobenius solutions for the equation: $8 x^{2} y^{\prime \prime}-2 x\left(3-4 x-x^{2}\right) y^{\prime}+\left(3+6 x+x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution.
|
$y_{1}=x^{3 / 2}\left(1-x+\frac{11}{26} x^{2}-\frac{109}{1326} x^{3}+\cdots\right)$
$y_{2}=x^{1 / 4}\left(1+4 x-\frac{131}{24} x^{2}+\frac{39}{14} x^{3}+\cdots\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{3 / 2}\left(1-x+\frac{11}{26} x^{2}-\frac{109}{1326} x^{3}+\cdots\right)$
$y_{2}=x^{1 / 4}\left(1+4 x-\frac{131}{24} x^{2}+\frac{39}{14} x^{3}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.7
|
Find the power series in $x$ for the general solution: $\left(1-x^{2}\right) y^{\prime \prime}-5 x y^{\prime}-4 y=0$
|
$y=a_{0} \sum_{m=0}^{\infty} \frac{2^{m} m !}{\prod_{j=0}^{m-1}(2 j+1)} x^{2 m}+a_{1} \sum_{m=0}^{\infty} \frac{\prod_{j=0}^{m-1}(2 j+3)}{2^{m} m !} x^{2 m+1}$
|
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} \frac{2^{m} m !}{\prod_{j=0}^{m-1}(2 j+1)} x^{2 m}+a_{1} \sum_{m=0}^{\infty} \frac{\prod_{j=0}^{m-1}(2 j+3)}{2^{m} m !} x^{2 m+1}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.34
|
Find a fundamental set of solutions: $\left(D^{4}-16\right)^{2} y=0$
|
$\left\{e^{2 x}, x e^{2 x}, e^{-2 x}, x e^{-2 x}, \cos 2 x, x \cos 2 x, \sin 2 x, x \sin 2 x\right\}$
|
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}, e^{-2 x}, x e^{-2 x}, \cos 2 x, x \cos 2 x, \sin 2 x, x \sin 2 x\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.2.7
|
A cup of boiling water is placed outside at 1:00 PM. One minute later the temperature of the water is $152^{\circ} \mathrm{F}$. After another minute its temperature is $112^{\circ} \mathrm{F}$. What is the outside temperature?
|
$32^{\circ} \mathrm{F}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
32^{\circ} \mathrm{F}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.15
|
Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-6 y=0$
|
$y=c_{1} x^{3}+\frac{c_{2}}{x^{2}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c_{1} x^{3}+\frac{c_{2}}{x^{2}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.4
|
A constant horizontal force of $10 \mathrm{~N}$ pushes a $20 \mathrm{~kg}$-mass through a medium that resists its motion with $.5 \mathrm{~N}$ for every $\mathrm{m} / \mathrm{s}$ of speed. The initial velocity of the mass is $7 \mathrm{~m} / \mathrm{s}$ in the direction opposite to the direction of the applied force. Find the velocity of the mass for $t>0$.
|
$v=20-27 e^{-t / 40}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
v=20-27 e^{-t / 40}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.4.4
|
An object with mass $m$ moves in a spiral orbit $r=c \theta^{2}$ under a central force
$$
\mathbf{F}(r, \theta)=f(r)(\cos \theta \mathbf{i}+\sin \theta \mathbf{j}) .
$$
Find $f$.
|
$f(r)=-m h^{2}\left(\frac{6 c}{r^{4}}+\frac{1}{r^{3}}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
f(r)=-m h^{2}\left(\frac{6 c}{r^{4}}+\frac{1}{r^{3}}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.49
|
Find two linearly independent Frobenius solutions of the equation: $x^{2}\left(1+x^{2}\right) y^{\prime \prime}-x\left(1-9 x^{2}\right) y^{\prime}+\left(1+25 x^{2}\right) y=0$
|
$y_{1}=x-4 x^{3}+x^{5}$
$y_{2}=y_{1} \ln x+6 x^{3}-3 x^{5}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x-4 x^{3}+x^{5}$
$y_{2}=y_{1} \ln x+6 x^{3}-3 x^{5}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.8
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-3 & 1 & -3 \\ 4 & -1 & 2 \\ 4 & -2 & 3\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1}\left[\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right] e^{t}+c_{2} e^{-t}\left[\begin{array}{c}-\sin 2 t-\cos 2 t \\ 2 \cos 2 t \\ 2 \cos 2 t\end{array}\right]+c_{3} e^{-t}\left[\begin{array}{c}\cos 2 t-\sin 2 t \\ 2 \sin 2 t \\ 2 \sin 2 t\end{array}\right]$
|
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}{r}-1 \\ 1 \\ 1\end{array}\right] e^{t}+c_{2} e^{-t}\left[\begin{array}{c}-\sin 2 t-\cos 2 t \\ 2 \cos 2 t \\ 2 \cos 2 t\end{array}\right]+c_{3} e^{-t}\left[\begin{array}{c}\cos 2 t-\sin 2 t \\ 2 \sin 2 t \\ 2 \sin 2 t\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.5.2
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{ll}0 & -1 \\ 1 & -2\end{array}\right] \mathbf{y}$
|
\mathbf{y}=c_{1}\left[\begin{array}{l}
1 \\
1
\end{array}\right] e^{-t}+c_{2}\left(\left[\begin{array}{l}
1 \\
0
\end{array}\right] e^{-t}+\left[\begin{array}{l}
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}{l}
1 \\
1
\end{array}\right] e^{-t}+c_{2}\left(\left[\begin{array}{l}
1 \\
0
\end{array}\right] e^{-t}+\left[\begin{array}{l}
1 \\
1
\end{array}\right] t e^{-t}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.4
|
Find the general solution: $y^{\prime \prime}-3 y^{\prime}+2 y=\frac{1}{1+e^{-x}} ; \quad y_{1}=e^{2 x}$
|
$y=\left(e^{2 x}+e^{x}\right) \ln \left(1+e^{-x}\right)+c_{1} e^{2 x}+c_{2} e^{x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\left(e^{2 x}+e^{x}\right) \ln \left(1+e^{-x}\right)+c_{1} e^{2 x}+c_{2} e^{x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.4.31
|
Solve the given homogeneous equation implicitly: $y^{\prime}=\frac{x+2 y}{2 x+y}$
|
$(y+x)=c(y-x)^{3} ; \quad y=x ; \quad y=-x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
(y+x)=c(y-x)^{3} ; \quad y=x ; \quad y=-x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.7.42
|
Find a fundamental set of Frobenius solutions of Bessel's equation:
$$
x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right) y=0
$$
in the case where $v$ is a positive integer.
|
$y_{1}=x^{\nu} \sum_{m=0}^{\infty} \frac{(-1)^{m}}{4^{m} m ! \prod_{j=1}^{m}(j+v)} x^{2 m}$;
$y_{2}=x^{-v} \sum_{m=0}^{v-1} \frac{(-1)^{m}}{4^{m} m ! \prod_{j=1}^{m}(j-v)} x^{2 m}-\frac{2}{4^{v} v !(v-1) !}\left(y_{1} \ln x-\frac{x^{v}}{2} \sum_{m=1}^{\infty} \frac{(-1)^{m}}{4^{m} m ! \prod_{j=1}^{m}(j+v)}\left(\sum_{j=1}^{m} \frac{2 j+v}{j(j+v)}\right) x^{2 m}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{\nu} \sum_{m=0}^{\infty} \frac{(-1)^{m}}{4^{m} m ! \prod_{j=1}^{m}(j+v)} x^{2 m}$;
$y_{2}=x^{-v} \sum_{m=0}^{v-1} \frac{(-1)^{m}}{4^{m} m ! \prod_{j=1}^{m}(j-v)} x^{2 m}-\frac{2}{4^{v} v !(v-1) !}\left(y_{1} \ln x-\frac{x^{v}}{2} \sum_{m=1}^{\infty} \frac{(-1)^{m}}{4^{m} m ! \prod_{j=1}^{m}(j+v)}\left(\sum_{j=1}^{m} \frac{2 j+v}{j(j+v)}\right) x^{2 m}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.1
|
Find the general solution: $y^{\prime}+a y=0(a=$ constant $)$
|
$y_{1}=x\left(1-x+\frac{3}{4} x^{2}-\frac{13}{36} x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+x^{2}\left(1-x+\frac{65}{108} x^{2}+\cdots\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x\left(1-x+\frac{3}{4} x^{2}-\frac{13}{36} x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+x^{2}\left(1-x+\frac{65}{108} x^{2}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.1.8
|
A weight stretches a spring 6 inches in equilibrium. The weight is initially displaced 6 inches above equilibrium and given a downward velocity of $3 \mathrm{ft} / \mathrm{s}$. Find its displacement for $t>0$.
|
$y=\frac{1}{2} \cos 8 t-\frac{3}{8} \sin 8 t \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=\frac{1}{2} \cos 8 t-\frac{3}{8} \sin 8 t \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.11
|
Find the general solution: $16 y^{(4)}-72 y^{\prime \prime}+81 y=0$
|
$y=e^{3 x / 2}\left(c_{1}+c_{2} x\right)+e^{-3 x / 2}\left(c_{3}+c_{4} x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{3 x / 2}\left(c_{1}+c_{2} x\right)+e^{-3 x / 2}\left(c_{3}+c_{4} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.45
|
Find two linearly independent Frobenius solutions of the equation: $x(1+x) y^{\prime \prime}+(1-x) y^{\prime}+y=0$
|
$y_{1}=1-x$
$y_{2}=y_{1} \ln x+4 x$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=1-x$
$y_{2}=y_{1} \ln x+4 x
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.2.7
|
Find the general solution: $27 y^{\prime \prime \prime}+27 y^{\prime \prime}+9 y^{\prime}+y=0$
|
$y=e^{-x / 3}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-x / 3}\left(c_{1}+c_{2} x+c_{3} x^{2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.15
|
Find the general solution: $x y^{\prime \prime}-(2 x+1) y^{\prime}+(x+1) y=-e^{x} ; \quad y_{1}=e^{x}$
|
$y=e^{x}\left(x+c_{1}+c_{2} x^{2}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{x}\left(x+c_{1}+c_{2} x^{2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.1.12
|
Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $y^{\prime \prime}-2 a y^{\prime}+a^{2} y=0$ $(a=$ constant $)$, given that $y_{1}=e^{a x}$.
|
$y_{2}=x e^{a x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{2}=x e^{a x}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.4.13
|
Find the general solution of the given Euler equation on $(0, \infty)$: $9 x^{2} y^{\prime \prime}+15 x y^{\prime}+y=0$
|
$y=x^{-1 / 3}\left(c_{1}+c_{2} \ln x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=x^{-1 / 3}\left(c_{1}+c_{2} \ln x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.2
|
Find the general solution: $y^{\prime}+3 x^{2} y=0$
|
$y_{1}=x^{-1}\left(1-2 x+\frac{9}{2} x^{2}-\frac{20}{3} x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+1-\frac{15}{4} x+\frac{133}{18} x^{2}+\cdots$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x^{-1}\left(1-2 x+\frac{9}{2} x^{2}-\frac{20}{3} x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+1-\frac{15}{4} x+\frac{133}{18} x^{2}+\cdots
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.23
|
Find a fundamental set of solutions: $x^{2} y^{\prime \prime}-(2 a-1) x y^{\prime}+a^{2} y=0 ; \quad y_{1}=x^{a}$
|
$\left\{x^{a}, x^{a} \ln x\right\}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
\left\{x^{a}, x^{a} \ln x\right\}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.2.1.2
|
Find the general solution: $y^{\prime}+3 x^{2} y=0$
|
$y=c e^{-x^{3}}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=c e^{-x^{3}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.10.6.7
|
Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}2 & 1 & -1 \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{array}\right] \mathbf{y}$
|
$\mathbf{y}=c_{1}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] e^{2 t}+c_{2} e^{t}\left[\begin{array}{r}
-\sin t \\
\sin t \\
\cos t
\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}
\cos t \\
-\cos t \\
\sin t
\end{array}\right]$
|
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}{l}
1 \\
1 \\
1
\end{array}\right] e^{2 t}+c_{2} e^{t}\left[\begin{array}{r}
-\sin t \\
\sin t \\
\cos t
\end{array}\right]+c_{3} e^{t}\left[\begin{array}{r}
\cos t \\
-\cos t \\
\sin t
\end{array}\right]
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.9.3.67
|
Find the general solution: $y^{\prime \prime \prime}-3 y^{\prime \prime}+4 y^{\prime \prime}-2 y^{\prime}=e^{x}[(28+6 x) \cos 2 x+(11-12 x) \sin 2 x]$
|
$y=-x e^{x} \sin 2 x+c_{1}+c_{2} e^{x}+e^{x}\left(c_{3} \cos x+c_{4} \sin x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-x e^{x} \sin 2 x+c_{1}+c_{2} e^{x}+e^{x}\left(c_{3} \cos x+c_{4} \sin x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.6.5
|
Find the general solution: $x^{2} y^{\prime}+y=0$
|
$y_{1}=x\left(1-4 x+\frac{19}{2} x^{2}-\frac{49}{3} x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+x^{2}\left(3-\frac{43}{4} x+\frac{208}{9} x^{2}+\cdots\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{1}=x\left(1-4 x+\frac{19}{2} x^{2}-\frac{49}{3} x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+x^{2}\left(3-\frac{43}{4} x+\frac{208}{9} x^{2}+\cdots\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.7
|
Find the general solution: $y^{\prime \prime}-8 y^{\prime}+16 y=0$
|
$y=e^{4 x}\left(c_{1}+c_{2} x\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{4 x}\left(c_{1}+c_{2} x\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.2.11
|
Find the general solution: $4 y^{\prime \prime}+4 y^{\prime}+10 y=0$
|
$y=e^{-x / 2}\left(c_{1} \cos \frac{3 x}{2}+c_{2} \sin \frac{3 x}{2}\right)$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-x / 2}\left(c_{1} \cos \frac{3 x}{2}+c_{2} \sin \frac{3 x}{2}\right)
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.17
|
A $192 \mathrm{lb}$ weight is suspended from a spring with constant $k=6 \mathrm{lb} / \mathrm{ft}$ and subjected to an external force $F(t)=8 \cos 3 t \mathrm{lb}$. Find the steady state component of the displacement for $t>0$ if the medium resists the motion with a force equal to 8 times the speed in $\mathrm{ft} / \mathrm{sec}$.
|
$y_{p}=-\frac{2}{15} \cos 3 t+\frac{1}{15} \sin 3 t \mathrm{ft}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{p}=-\frac{2}{15} \cos 3 t+\frac{1}{15} \sin 3 t \mathrm{ft}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.7.2.5
|
Find the power series in $x$ for the general solution: $\left(1+2 x^{2}\right) y^{\prime \prime}+7 x y^{\prime}+2 y=0$
|
$y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} \frac{4 j+1}{2 j+1}\right] x^{2 m}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}(4 j+3)\right] \frac{x^{2 m+1}}{2^{m} m !}$
|
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}\left[\prod_{j=0}^{m-1} \frac{4 j+1}{2 j+1}\right] x^{2 m}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}(4 j+3)\right] \frac{x^{2 m+1}}{2^{m} m !}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.4.3.11
|
An object with mass $m$ is given an initial velocity $v_{0} \leq 0$ in a medium that exerts a resistive force with magnitude proportional to the square of the speed. Find the velocity of the object for $t>0$, and find its terminal velocity.
|
$v=\alpha \frac{v_{0}\left(1+e^{-\beta t}\right)-\alpha\left(1-e^{-\beta t}\right)}{\alpha\left(1+e^{-\beta t}\right)-v_{0}\left(1-e^{-\beta t}\right)} ; \quad-\alpha$, where $\alpha=\sqrt{\frac{m g}{k}}$ and $\beta=2 \sqrt{\frac{k g}{m}}$.
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
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college_math.differential_equation
|
v=\alpha \frac{v_{0}\left(1+e^{-\beta t}\right)-\alpha\left(1-e^{-\beta t}\right)}{\alpha\left(1+e^{-\beta t}\right)-v_{0}\left(1-e^{-\beta t}\right)} ; \quad-\alpha$, where $\alpha=\sqrt{\frac{m g}{k}}$ and $\beta=2 \sqrt{\frac{k g}{m}}$.
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college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
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exercise.6.4.5
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An object with mass $m$ moves in the orbit $r=r_{0} e^{\gamma \theta}$ under a central force
$$
\mathbf{F}(r, \theta)=f(r)(\cos \theta \mathbf{i}+\sin \theta \mathbf{j}) .
$$
Find $f$.
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$f(r)=-\frac{m h^{2}\left(\gamma^{2}+1\right)}{r^{3}}$
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
f(r)=-\frac{m h^{2}\left(\gamma^{2}+1\right)}{r^{3}}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.6.2.15
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A mass of one $\mathrm{kg}$ stretches a spring $49 \mathrm{~cm}$ in equilibrium. A dashpot attached to the spring supplies a damping force of $4 \mathrm{~N}$ for each $\mathrm{m} / \mathrm{sec}$ of speed. The mass is initially displaced $10 \mathrm{~cm}$ above equilibrium and given a downward velocity of $1 \mathrm{~m} / \mathrm{sec}$. Find its displacement for $t>0$.
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$y=e^{-2 t}\left(\frac{1}{10} \cos 4 t-\frac{1}{5} \sin 4 t\right) \mathrm{m}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=e^{-2 t}\left(\frac{1}{10} \cos 4 t-\frac{1}{5} \sin 4 t\right) \mathrm{m}
|
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.1.13
|
Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $x^{2} y^{\prime \prime}+x y^{\prime}-y=0$, given that $y_{1}=x$.
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$y_{2}=\frac{1}{x}$
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y_{2}=\frac{1}{x}
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college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS
|
exercise.5.6.12
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Find the general solution: $(1-2 x) y^{\prime \prime}+2 y^{\prime}+(2 x-3) y=\left(1-4 x+4 x^{2}\right) e^{x} ; \quad y_{1}=e^{x}$
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$y=-\frac{(2 x-1)^{2} e^{x}}{8}+c_{1} e^{x}+c_{2} x e^{-x}$
|
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0)
|
college_math.differential_equation
|
y=-\frac{(2 x-1)^{2} e^{x}}{8}+c_{1} e^{x}+c_{2} x e^{-x}
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