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college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.10 | A $32 \mathrm{lb}$ weight stretches a spring $1 \mathrm{ft}$ in equilibrium. The weight is initially displaced 6 inches above equilibrium and given a downward velocity of $3 \mathrm{ft} / \mathrm{sec}$. Find its displacement for $t>0$ if the medium resists the motion with a force equal to 3 times the speed in $\mathrm{ft} / \mathrm{sec}$. | $y=e^{-\frac{3}{2} t}\left(\frac{1}{2} \cos \frac{\sqrt{119}}{2} t-\frac{9}{2 \sqrt{119}} \sin \frac{\sqrt{119}}{2} t\right) \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.1.17 | Suppose $y(x)=\sum_{n=0}^{\infty} a_{n}(x-2)^{n}$ on an open interval that contains $x_{0}=2$. Find a power series in $x-2$ for $x^{2} y^{\prime \prime}+2 x y^{\prime}-3 x y$. | $b_{0}=8 a_{2}+4 a_{1}-6 a_{0}$,
$b_{n}=4(n+2)(n+1) a_{n+2}+4(n+1)^{2} a_{n+1}+\left(n^{2}+n-6\right) a_{n}-3 a_{n-1}, n \geq 1$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.4.19 | Find the general solution for the equation: $y^{\prime \prime}-2 y^{\prime}+y=e^{x}(2-12 x)$ | $y=e^{x}\left[x^{2}(1-2 x)+c_{1}+c_{2} x\right]$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.6 | Find the general solution: $4 x^{2} y^{\prime \prime}+\left(4 x-8 x^{2}\right) y^{\prime}+\left(4 x^{2}-4 x-1\right) y=4 x^{1 / 2} e^{x}(1+4 x) ; \quad y_{1}=x^{1 / 2} e^{x}$ | $y=e^{x}\left(2 x^{3 / 2}+x^{1 / 2} \ln x+c_{1} x^{1 / 2}+c_{2} x^{-1 / 2}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.9 | A $64 \mathrm{lb}$ weight is suspended from a spring with constant $k=25 \mathrm{lb} / \mathrm{ft}$. It is initially displaced 18 inches above equilibrium and released from rest. Find its displacement for $t>0$ if the medium resists the motion with $6 \mathrm{lb}$ of force for each $\mathrm{ft} / \mathrm{sec}$ of velocity. | $y=e^{-3 t / 2}\left(\frac{3}{2} \cos \frac{\sqrt{41}}{2} t+\frac{9}{2 \sqrt{41}} \sin \frac{\sqrt{41}}{2} t\right) \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.6.50 | Find two linearly independent Frobenius solutions of the equation: $9 x^{2} y^{\prime \prime}+3 x\left(1-x^{2}\right) y^{\prime}+\left(1+7 x^{2}\right) y=0$ | $y_{1}=x^{1 / 3}\left(1-\frac{1}{6} x^{2}\right)$
$y_{2}=y_{1} \ln x+x^{7 / 3}\left(\frac{1}{4}-\frac{1}{12} \sum_{m=1}^{\infty} \frac{1}{6^{m} m(m+1)(m+1) !} x^{2 m}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.6.44 | Find two linearly independent Frobenius solutions of the equation: $x^{2}(1-2 x) y^{\prime \prime}+3 x y^{\prime}+(1+4 x) y=0$ | $y_{1}=\frac{1}{x}$
$y_{2}=y_{1} \ln x-6+6 x-\frac{8}{3} x^{2}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.34 | Solve the given homogeneous equation implicitly: $y^{\prime}=\frac{x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}}$ | $\frac{y}{x}+\frac{y^{3}}{x^{3}}=\ln |x|+c$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.6.15 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}6 & 0 & -3 \\ -3 & 3 & 3 \\ 1 & -2 & 6\end{array}\right] \mathbf{y}^{\prime}$ | $\mathbf{y}=c_{1}\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right] e^{3 t}+c_{2} e^{6 t}\left[\begin{array}{r}-\sin 3 t \\ \sin 3 t \\ \cos 3 t\end{array}\right]+c_{3} e^{6 t}\left[\begin{array}{r}\cos 3 t \\ -\cos 3 t \\ \sin 3 t\end{array}\right]$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.1.21 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $\left(x^{2}-4\right) y^{\prime \prime}+4 x y^{\prime}+2 y=0$, given that $y_{1}=\frac{1}{x-2}$. | $y_{2}=\frac{1}{x^{2}-4}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.27 | Find a fundamental set of solutions: $4 x^{2} y^{\prime \prime}-4 x y^{\prime}+\left(3-16 x^{2}\right) y=0 ; \quad y_{1}=x^{1 / 2} e^{2 x}$ | $\left\{x^{1 / 2} e^{2 x}, x^{1 / 2} e^{-2 x}\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.29 | Find a fundamental set of solutions: $\left(D^{2}+6 D+13\right)(D-2)^{2} D^{3} y=0$ | $\left\{e^{-3 x} \cos 2 x, e^{-3 x} \sin 2 x, e^{2 x}, x e^{2 x}, 1, x, x^{2}\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.2.8 | Find the general solution: $y^{\prime \prime}+y^{\prime}=0$ | $y=c_{1}+c_{2} e^{-x}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.30 | Find a fundamental set of solutions: $x y^{\prime \prime}-(4 x+1) y^{\prime}+(4 x+2) y=0 ; \quad y_{1}=e^{2 x}$ | $\left\{e^{2 x}, x^{2} e^{2 x}\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.3.9 | Find the steady state current in the circuit described by the equation.
$\frac{1}{10} Q^{\prime \prime}+6 Q^{\prime}+250 Q=10 \cos 100 t+30 \sin 100 t$ | $I_{p}=\frac{20}{123}(17 \sin 100 t-11 \cos 100 t)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.5.5 | Find a fundamental set of Frobenius solutions for the equation: $12 x^{2}(1+x) y^{\prime \prime}+x\left(11+35 x+3 x^{2}\right) y^{\prime}-\left(1-10 x-5 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution. | $y_{1}=x^{1 / 3}\left(1-x+\frac{28}{31} x^{2}-\frac{1111}{1333} x^{3}+\cdots\right)$
$y_{2}=x^{1 / 4}\left(1-x+\frac{7}{8} x^{2}-\frac{19}{24} x^{3}+\cdots\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.8.7.29 | Given the equation $m y^{\prime \prime}+c y^{\prime}+k y=0, \quad y(0)=y_{0}, \quad y^{\prime}(0)=v_{0}$, find the impulse that would have to be applied to the object at $t=\tau$ to put it in equilibrium if $y(\tau)=0$. | $y=(-1)^{k} m \omega_{1} R e^{-c \tau / 2 m} \delta(t-\tau)$ if $\omega_{1} \tau-\phi=(2 k+1) \pi / 2(k=$ integer) | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.8 | Find the general solution: $y^{\prime \prime}+4 x y^{\prime}+\left(4 x^{2}+2\right) y=8 e^{-x(x+2)} ; \quad y_{1}=e^{-x^{2}}$ | $y=e^{-x^{2}}\left(2 e^{-2 x}+c_{1}+c_{2} x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.1 | Solve the given Bernoulli equation: $y^{\prime}+y=y^{2}$ | $y=\frac{1}{1-c e^{x}}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.37 | Find a fundamental set of solutions: $\left(4 D^{2}+1\right)^{2}\left(9 D^{2}+4\right)^{3} y=0$ | $\{\cos (x / 2), x \cos (x / 2), \sin (x / 2), x \sin (x / 2), \cos 2 x / 3 x \cos (2 x / 3)$, $\left.x^{2} \cos (2 x / 3), \sin (2 x / 3), x \sin (2 x / 3), x^{2} \sin (2 x / 3)\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.4.16 | Find the general solution of the given Euler equation on $(0, \infty)$: $2 x^{2} y^{\prime \prime}+3 x y^{\prime}-y=0$ | $y=\frac{c_{1}}{x}+c_{2} x^{1 / 2}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.5.26 | Find the orthogonal trajectories of the given family of curves: $x^{2}+4 x y+y^{2}=c$ | $(y-x)^{3}(y+x)=k$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.2.12 | Suppose water is added to a tank at $10 \mathrm{gal} / \mathrm{min}$, but leaks out at the rate of $1 / 5 \mathrm{gal} / \mathrm{min}$ for each gallon in the tank. What is the smallest capacity the tank can have if the process is to continue indefinitely? | 50 gallons | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.3.6 | Find the steady state current in the circuit described by the equation.
$\frac{1}{10} Q^{\prime \prime}+3 Q^{\prime}+100 Q=5 \cos 10 t-5 \sin 10 t$ | $I_{p}=-\frac{1}{3}(\cos 10 t+2 \sin 10 t)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.5.8 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}0 & 2 & 1 \\ -4 & 6 & 1 \\ 0 & 4 & 2\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
-1 \\
2
\end{array}\right]+c_{2}\left[\begin{array}{l}
1 \\
1 \\
2
\end{array}\right] e^{4 t}+c_{3}\left(\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right] \frac{e^{4 t}}{2}+\left[\begin{array}{l}
1 \\
1 \\
2
\end{array}\right] t e^{4 t}\right) | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.3.10 | Find the steady state current in the circuit described by the equation.
$\frac{1}{20} Q^{\prime \prime}+4 Q^{\prime}+125 Q=15 \cos 30 t-30 \sin 30 t$ | $I_{p}=-\frac{45}{52}(\cos 30 t+8 \sin 30 t)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.15 | Solve the equation explicitly: $y^{\prime}=\frac{y+x}{x}$ | $y=x(\ln |x|+c)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.5.25 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}1 & 10 & -12 \\ 2 & 2 & 3 \\ 2 & -1 & 6\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right] e^{3 t}+c_{2}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{3 t}}{2}+\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right] t e^{3 t}\right)+c_{3}\left(\left[\begin{array}{l}
1 \\
2 \\
0
\end{array}\right] \frac{e^{3 t}}{36}+\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{t e^{3 t}}{2}+\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right] \frac{t^{2} e^{3 t}}{2}\right) | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.18 | A 2 gm mass is attached to a spring with constant 20 dyne/cm. Find the steady state component of the displacement if the mass is subjected to an external force $F(t)=3 \cos 4 t-5 \sin 4 t$ dynes and a dashpot supplies 4 dynes of damping for each $\mathrm{cm} / \mathrm{sec}$ of velocity. | $y_{p}=\frac{11}{100} \cos 4 t+\frac{27}{100} \sin 4 t \mathrm{~cm}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.16 | A mass of 100 grams stretches a spring $98 \mathrm{~cm}$ in equilibrium. A dashpot attached to the spring supplies a damping force of 600 dynes for each $\mathrm{cm} / \mathrm{sec}$ 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 $t>0$. | $y=e^{-3 t}(10 \cos t-70 \sin t) \mathrm{cm}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.1.13 | Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $\left(1+2 x^{2}\right) y^{\prime \prime}+(2-3 x) y^{\prime}+4 y$. | $b_{n}=(n+2)(n+1) a_{n+2}+2(n+1) a_{n+1}+\left(2 n^{2}-5 n+4\right) a_{n}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.2.19 | Identical tanks $T_{1}$ and $T_{2}$ initially contain $W$ gallons each of pure water. Starting at $t_{0}=0$, a salt solution with constant concentration $c$ is pumped into $T_{1}$ at $r \mathrm{gal} / \mathrm{min}$ and drained from $T_{1}$ into $T_{2}$ at the same rate. The resulting mixture in $T_{2}$ is also drained at the same rate. Find the concentrations $c_{1}(t)$ and $c_{2}(t)$ in tanks $T_{1}$ and $T_{2}$ for $t>0$. | $c_{1}=c\left(1-e^{-r t / W}\right), c_{2}=c\left(1-e^{-r t / W}-\frac{r}{W} t e^{-r t / W}\right)$. | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.10 | A $64 \mathrm{lb}$ weight is attached to a spring with constant $k=8 \mathrm{lb} / \mathrm{ft}$ and subjected to an external force $F(t)=2 \sin t$. The weight is initially displaced 6 inches above equilibrium and given an upward velocity of $2 \mathrm{ft} / \mathrm{s}$. Find its displacement for $t>0$. | $y=\frac{1}{3} \sin t+\frac{1}{2} \cos 2 t+\frac{5}{6} \sin 2 t$ ft | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.3 | Solve the given Bernoulli equation: $x^{2} y^{\prime}+2 y=2 e^{1 / x} y^{1 / 2}$ | $y=e^{2 / x}(c-1 / x)^{2}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.5 | Find the general solution: $y^{\prime \prime \prime}+5 y^{\prime \prime}+9 y^{\prime}+5 y=0$ | $y=c_{1} e^{-x}+e^{-2 x}\left(c_{1} \cos x+c_{2} \sin x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.8.4.29 | Find $L(u(t-\tau))$. | $\frac{e^{-\tau s}}{s}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.26 | Find a fundamental set of solutions: $4 x^{2}(\sin x) y^{\prime \prime}-4 x(x \cos x+\sin x) y^{\prime}+(2 x \cos x+3 \sin x) y=0 ; \quad y_{1}=x^{1 / 2}$ | $\left\{x^{1 / 2}, x^{1 / 2} \cos x\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.5.33 | Find conditions on the constants $A, B, C$, and $D$ such that the equation
$(A x+B y) d x+(C x+D y) d y=0$
is exact. | $B=C$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.5.9 | Find a fundamental set of Frobenius solutions for the equation: $x\left(3+x+x^{2}\right) y^{\prime \prime}+\left(4+x-x^{2}\right) y^{\prime}+x y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution. | $y_{1}=1-\frac{1}{14} x^{2}+\frac{1}{105} x^{3}+\cdots$
$y_{2}=x^{-1 / 3}\left(1-\frac{1}{18} x-\frac{71}{405} x^{2}+\frac{719}{34992} x^{3}+\cdots\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.1.22 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $(2 x+1) x y^{\prime \prime}-2\left(2 x^{2}-1\right) y^{\prime}-4(x+1) y=0$, given that $y_{1}=\frac{1}{x}$. | $y_{2}=e^{2 x}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.5.3 | Find a fundamental set of Frobenius solutions for the equation: $x^{2}\left(3+3 x+x^{2}\right) y^{\prime \prime}+x\left(5+8 x+7 x^{2}\right) y^{\prime}-\left(1-2 x-9 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution. | $y_{1}=x^{1 / 3}\left(1-\frac{4}{7} x-\frac{7}{45} x^{2}+\frac{970}{2457} x^{3}+\cdots\right)$
$y_{2}=x^{-1}\left(1-x^{2}+\frac{2}{3} x^{3}+\cdots\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.29 | Solve the given homogeneous equation implicitly: $\left(y^{\prime} x-y\right)(\ln |y|-\ln |x|)=x$ | $(x+y) \ln |x|+y(1-\ln |y|)+c x=0$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.5 | A $16 \mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. It is attached to a damping mechanism with constant $c$. Find all values of $c$ such that the free vibration of the weight has infinitely many oscillations. | $0 \leq c<8 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.4.18 | Find the general solution for the equation: $y^{\prime \prime}+2 y^{\prime}-3 y=-16 x e^{x}$ | $y=x e^{x}(1-2 x)+c_{1} e^{x}+c_{2} e^{-3 x}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.1.43 | Experiments indicate that glucose is absorbed by the body at a rate proportional to the amount of glucose present in the bloodstream. Let $\lambda$ denote the (positive) constant of proportionality. Now suppose glucose is injected into a patient's bloodstream at a constant rate of $r$ units per unit of time. Let $G=G(t)$ be the number of units in the patient's bloodstream at time $t>0$. Then
$$
G^{\prime}=-\lambda G+r
$$
where the first term on the right is due to the absorption of the glucose by the patient's body and the second term is due to the injection. Determine $G$ for $t>0$, given that $G(0)=G_{0}$. Also, find $\lim _{t \rightarrow \infty} G(t)$. | $G=\frac{r}{\lambda}+\left(G_{0}-\frac{r}{\lambda}\right) e^{-\lambda t}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.5.6 | Find a fundamental set of Frobenius solutions for the equation: $x^{2}\left(5+x+10 x^{2}\right) y^{\prime \prime}+x\left(4+3 x+48 x^{2}\right) y^{\prime}+\left(x+36 x^{2}\right) y=0$. Compute $a_{0}, a_{1} \ldots, a_{N}$ for $N$ at least 7 in each solution. | $y_{1}=x^{1 / 5}\left(1-\frac{6}{25} x-\frac{1217}{625} x^{2}+\frac{41972}{46875} x^{3}+\cdots\right)$
$y_{2}=x-\frac{1}{4} x^{2}-\frac{35}{18} x^{3}+\frac{11}{12} x^{4}+\cdots$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.18 | Find a fundamental set of solutions: $x y^{\prime \prime}+(2-2 x) y^{\prime}+(x-2) y=0 ; \quad y_{1}=e^{x}$ | $\{e^{x}, e^{x} / x\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.5.29 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-1 & -12 & 8 \\ 1 & -9 & 4 \\ 1 & -6 & 1\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{r}
-4 \\
0 \\
1
\end{array}\right] e^{-3 t}+c_{2}\left[\begin{array}{l}
6 \\
1 \\
0
\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}
2 \\
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 |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.28 | Find a fundamental set of solutions: $(2 x+1) x y^{\prime \prime}-2\left(2 x^{2}-1\right) y^{\prime}-4(x+1) y=0 ; \quad y_{1}=1 / x$ | $\left\{1 / x, e^{2 x}\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.6 | Find the general solution: $4 y^{\prime \prime \prime}-8 y^{\prime \prime}+5 y^{\prime}-y=0$ | $y=c_{1} e^{x}+e^{x / 2}\left(c_{2}+c_{3} x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.4.12 | Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}+3 x y^{\prime}+5 y=0$ | $y=\frac{1}{x}\left[c_{1} \cos (2 \ln x)+c_{2} \sin (2 \ln x]\right.$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.15 | A $6 \mathrm{lb}$ weight stretches a spring 6 inches in equilibrium. Suppose an external force $F(t)=$ $\frac{3}{16} \sin \omega t+\frac{3}{8} \cos \omega t \mathrm{lb}$ is applied to the weight. For what value of $\omega$ will the displacement be unbounded? Find the displacement if $\omega$ has this value. Assume that the motion starts from equilibrium with zero initial velocity. | $\omega=8 \mathrm{rad} / \mathrm{s} y=-\frac{t}{16}(-\cos 8 t+2 \sin 8 t)+\frac{1}{128} \sin 8 t \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.5.1 | Determine which equations are exact and solve them:
1. $6 x^{2} y^{2} d x+4 x^{3} y d y=0$
2. $\left(3 y \cos x+4 x e^{x}+2 x^{2} e^{x}\right) d x+(3 \sin x+3) d y=0$
3. $14 x^{2} y^{3} d x+21 x^{2} y^{2} d y=0$
4. $\left(2 x-2 y^{2}\right) d x+\left(12 y^{2}-4 x y\right) d y=0$
5. $(x+y)^{2} d x+(x+y)^{2} d y=0$
6. $(4 x+7 y) d x+(3 x+4 y) d y=0$
7. $\left(-2 y^{2} \sin x+3 y^{3}-2 x\right) d x+\left(4 y \cos x+9 x y^{2}\right) d y=0$
8. $(2 x+y) d x+(2 y+2 x) d y=0$
9. $\left(3 x^{2}+2 x y+4 y^{2}\right) d x+\left(x^{2}+8 x y+18 y\right) d y=0$
10. $\left(2 x^{2}+8 x y+y^{2}\right) d x+\left(2 x^{2}+x y^{3} / 3\right) d y=0$
11. $\left(\frac{1}{x}+2 x\right) d x+\left(\frac{1}{y}+2 y\right) d y=0$
12. $\left(y \sin x y+x y^{2} \cos x y\right) d x+\left(x \sin x y+x y^{2} \cos x y\right) d y=0$
13. $\frac{x d x}{\left(x^{2}+y^{2}\right)^{3 / 2}}+\frac{y d y}{\left(x^{2}+y^{2}\right)^{3 / 2}}=0$
14. $\left(e^{x}\left(x^{2} y^{2}+2 x y^{2}\right)+6 x\right) d x+\left(2 x^{2} y e^{x}+2\right) d y=0$
15. $\left(x^{2} e^{x^{2}+y}\left(2 x^{2}+3\right)+4 x\right) d x+\left(x^{3} e^{x^{2}+y}-12 y^{2}\right) d y=0$
16. $\left(e^{x y}\left(x^{4} y+4 x^{3}\right)+3 y\right) d x+\left(x^{5} e^{x y}+3 x\right) d y=0$
17. $\left(3 x^{2} \cos x y-x^{3} y \sin x y+4 x\right) d x+\left(8 y-x^{4} \sin x y\right) d y=0$ | $2 x^{3} y^{2}=c$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.6 | A $10 \mathrm{~kg}$ mass stretches a spring $70 \mathrm{~cm}$ in equilibrium. Suppose a $2 \mathrm{~kg}$ mass is attached to the spring, initially displaced $25 \mathrm{~cm}$ below equilibrium, and given an upward velocity of $2 \mathrm{~m} / \mathrm{s}$. Find its displacement for $t>0$. Find the frequency, period, amplitude, and phase angle of the motion. | $y=-\frac{1}{4} \cos \sqrt{70} t+\frac{2}{\sqrt{70}} \sin \sqrt{70} t \mathrm{~m} ; \quad R=\frac{1}{4} \sqrt{\frac{67}{35}} \mathrm{~m} \omega_{0}=\sqrt{70} \mathrm{rad} / \mathrm{s}$;
$T=2 \pi / \sqrt{70} \mathrm{~s} ; \phi \approx 2.38 \mathrm{rad} \approx 136.28^{\circ}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.2.38 | Solve the equation using variation of parameters followed by separation of variables: $y^{\prime}-2 y=\frac{x e^{2 x}}{1-y e^{-2 x}}$ | $y=e^{2 x}\left(1 \pm \sqrt{c-x^{2}}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.3.7 | A $96 \mathrm{lb}$ weight is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the speed. Find its velocity as a function of time if its terminal velocity is $-128 \mathrm{ft} / \mathrm{s}$. | $v=-128\left(1-e^{-t / 4}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.3.18 | A space vehicle is to be launched from the moon, which has a radius of about 1080 miles. The acceleration due to gravity at the surface of the moon is about $5.31 \mathrm{ft} / \mathrm{s}^{2}$. Find the escape velocity in miles/s. | $\approx 1.47$ miles/s | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.1.4 | Find the general solution: $x y^{\prime}+3 y=0$ | $y=\frac{c}{x^{3}}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.3.17 | A space probe is to be launched from a space station 200 miles above Earth. Determine its escape velocity in miles/s. Take Earth's radius to be 3960 miles. | $\approx 6.76$ miles/s | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.1.11 | Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $(2+x) y^{\prime \prime}+x y^{\prime}+3 y$. | $b_{n}=2(n+2)(n+1) a_{n+2}+(n+1) n a_{n+1}+(n+3) a_{n}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.25 | Find a fundamental set of solutions: $x y^{\prime \prime}-(4 x+1) y^{\prime}+(4 x+2) y=0 ; \quad y_{1}=e^{2 x}$ | $\left\{e^{2 x}, x^{2} e^{2 x}\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.6.4 | Find the general solution: $x y^{\prime}+3 y=0$ | $y_{1}=x^{1 / 2}\left(1-2 x+\frac{5}{2} x^{2}-2 x^{3}+\cdots\right)$
$y_{2}=y_{1} \ln x+x^{3 / 2}\left(1-\frac{9}{4} x+\frac{17}{6} x^{2}+\cdots\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.3.63 | Find the general solution: $y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=-2 e^{-x}\left(7-18 x+6 x^{2}\right)$ | $y=x^{2} e^{-x}(1-x)^{2}+c_{1}+e^{-x}\left(c_{2}+c_{3} x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.5.21 | Find a curve $y=y(x)$ through $(1,-1)$ such that the tangent to the curve at any point $\left(x_{0}, y\left(x_{0}\right)\right)$ intersects the $y$ axis at $y_{I}=x_{0}^{3}$. | $y=-\frac{x^{3}}{2}-\frac{x}{2}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.2.36 | Solve the equation using variation of parameters followed by separation of variables: $x y^{\prime}-2 y=\frac{x^{6}}{y+x^{2}}$ | $y=x^{2}\left(-1+\sqrt{x^{2}+c}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.33 | Find a fundamental set of solutions: $\left(D^{2}+1\right)\left(D^{2}+9\right)^{2}(D-2) y=0$ | $\left\{\cos x, \sin x, \cos 3 x, x \cos 3 x, \sin 3 x, x \sin 3 x, e^{2 x}\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.6.48 | Find two linearly independent Frobenius solutions of the equation: $x^{2}(1-x) y^{\prime \prime}-x(3-5 x) y^{\prime}+(4-5 x) y=0$ | $y_{1}=x^{2}(1-x)^{3}$
$y_{2}=y_{1} \ln x+x^{3}\left(4-7 x+\frac{11}{3} x^{2}-6 \sum_{n=3}^{\infty} \frac{1}{n(n-2)\left(n^{2}-1\right)} x^{n}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.11 | Find the general solution: $x^{2} y^{\prime \prime}-x(2 x-1) y^{\prime}+\left(x^{2}-x-1\right) y=x^{2} e^{x} ; \quad y_{1}=x e^{x}$ | $y=x e^{x}\left(\frac{x}{3}+c_{1}+\frac{c_{2}}{x^{2}}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.1.20 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $(3 x-1) y^{\prime \prime}-(3 x+2) y^{\prime}-(6 x-8) y=0$, given that $y_{1}=e^{2 x}$. | $y_{2}=x e^{-x}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.6.2 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{ll}-11 & 4 \\ -26 & 9\end{array}\right] \mathbf{y}$ | $\mathbf{y}=c_{1} e^{-t}\left[\begin{array}{c}
5 \cos 2 t+\sin 2 t \\
13 \cos 2 t
\end{array}\right]+c_{2} e^{-t}\left[\begin{array}{c}
5 \sin 2 t-\cos 2 t \\
13 \sin 2 t
\end{array}\right]$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.6.14 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}3 & -4 & -2 \\ -5 & 7 & -8 \\ -10 & 13 & -8\end{array}\right] \mathbf{y}$ | $\mathbf{y}=c_{1}\left[\begin{array}{l}2 \\ 2 \\ 1\end{array}\right] e^{-2 t}+c_{2} e^{2 t}\left[\begin{array}{c}-\cos 3 t-\sin 3 t \\ -\sin 3 t \\ \cos 3 t\end{array}\right]+c_{3} e^{2 t}\left[\begin{array}{c}-\sin 3 t+\cos 3 t \\ \cos 3 t \\ \sin 3 t\end{array}\right]$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.5.22 | 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 $y$ axis at $y_{I}=x_{0}$. | $y=-x \ln |x|+c x$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.4 | A $96 \mathrm{lb}$ weight stretches a spring $3.2 \mathrm{ft}$ in equilibrium. It is attached to a dashpot with damping constant $c=18 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}$. The weight is initially displaced 15 inches below equilibrium and given a downward velocity of $12 \mathrm{ft} / \mathrm{sec}$. Find its displacement for $t>0$. | $y=-\frac{e^{-3 t}}{4}(5 \cos t+63 \sin t) \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.13 | Find the general solution: $x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=4 x^{4} ; \quad y_{1}=x^{2}$ | $y=x^{4}+c_{1} x^{2}+c_{2} x^{2} \ln |x|$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.1.3 | Find the general solution: $x y^{\prime}+(\ln x) y=0$ | $y=c e^{-(\ln x)^{2} / 2}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.13 | An $8 \mathrm{lb}$ weight stretches a spring 8 inches in equilibrium. It is attached to a dashpot with damping constant $c=.5 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}$ and subjected to an external force $F(t)=4 \cos 2 t \mathrm{lb}$. Determine the steady state component of the displacement for $t>0$. | $y_{p}=\frac{22}{61} \cos 2 t+\frac{2}{61} \sin 2 t \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.30 | Solve the given homogeneous equation implicitly: $y^{\prime}=\frac{y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x(y+x)^{2}}$ | $(y+x)^{3}=3 x^{3}(\ln |x|+c)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.4.8 | Find the general solution of the given Euler equation on $(0, \infty)$: $12 x^{2} y^{\prime \prime}-5 x y^{\prime \prime}+6 y=0$ | $y=c_{1} x^{2 / 3}+c_{2} x^{3 / 4}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.6.1 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{ll}-1 & 2 \\ -5 & 5\end{array}\right] \mathbf{y}$ | $\mathbf{y}=c_{1} e^{2 t}\left[\begin{array}{c}
3 \cos t+\sin t \\
5 \cos t
\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{c}
3 \sin t-\cos t \\
5 \sin t
\end{array}\right]$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.4.17 | Find the general solution for the equation: $y^{\prime \prime}+6 y^{\prime}+9 y=e^{2 x}(3-5 x)$ | $y=\frac{e^{2 x}}{5}(1-x)+e^{-3 x}\left(c_{1}+c_{2} x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.11 | A unit mass hangs in equilibrium from a spring with constant $k=1 / 16$. Starting at $t=0$, a force $F(t)=3 \sin t$ is applied to the mass. Find its displacement for $t>0$. | $y=\frac{16}{5}\left(4 \sin \frac{t}{4}-\sin t\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.21 | Two identical objects suspended from different springs are set into motion. The period of one motion is 3 times the period of the other. How are the two spring constants related? | $k_{1}=9 k_{2}$, where $k_{1}$ is the spring constant of the system with the shorter period. | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.2.28 | The population $P=P(t)$ of a species satisfies the logistic equation: $P^{\prime}=a P(1-\alpha P)$ and $P(0)=P_{0}>0$. Find $P$ for $t>0$, and find $\lim _{t \rightarrow \infty} P(t)$. | $P=\frac{P_{0}}{\alpha P_{0}+\left(1-\alpha P_{0}\right) e^{-a t}} ; \lim _{t \rightarrow \infty} P(t)=1 / \alpha$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.2.14 | A $32 \mathrm{lb}$ weight stretches a spring $1 \mathrm{ft}$ in equilibrium. It is attached to a dashpot with constant $c=12 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}$. The weight is initially displaced 8 inches above equilibrium and released from rest. Find its displacement for $t>0$. | $y=-\frac{2}{3}\left(e^{-8 t}-2 e^{-4 t}\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.35 | Find a fundamental set of solutions: $\left(4 D^{2}+4 D+9\right)^{3} y=0$ | $\left\{e^{-x / 2} \cos 2 x, x e^{-x / 2} \cos 2 x, x^{2} e^{-x / 2} \cos 2 x, e^{-x / 2} \sin 2 x, x e^{-x / 2} \sin 2 x\right.$, $\left.x^{2} e^{-x / 2} \sin 2 x\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.3.62 | Find the general solution: $y^{\prime \prime \prime}-6 y^{\prime \prime}+11 y^{\prime}-6 y=e^{2 x}\left(5-4 x-3 x^{2}\right)$ | $y=x e^{2 x}(1+x)^{2}+c_{1} e^{x}+c_{2} e^{2 x}+c_{3} e^{3 x}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.7 | A weight stretches a spring 1.5 inches in equilibrium. The weight is initially displaced 8 inches above equilibrium and given a downward velocity of $4 \mathrm{ft} / \mathrm{s}$. Find its displacement for $t>0$. | $y=\frac{2}{3} \cos 16 t-\frac{1}{4} \sin 16 t \mathrm{ft}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.1.14 | 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 $y_{1}=x$. | $y_{2}=x \ln x$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.2.11 | A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with $1 / 4$ pound of salt per gallon is added to the tank at $4 \mathrm{gal} / \mathrm{min}$, and the resulting mixture is drained out at $2 \mathrm{gal} / \mathrm{min}$. Find the quantity of salt in the tank as it's about to overflow. | $Q(50)=47.5$ (pounds) | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.10 | Find the general solution: $y^{(4)}+12 y^{\prime \prime}+36 y=0$ | $y=\left(c_{1}+c_{2} x\right) \cos \sqrt{6} x+\left(c_{3}+c_{4} x\right) \sin \sqrt{6} x$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.6.1.9 | A spring-mass system has natural frequency $7 \sqrt{10} \mathrm{rad} / \mathrm{s}$. The natural length of the spring is $.7 \mathrm{~m}$. What is the length of the spring when the mass is in equilibrium? | $.72 \mathrm{~m}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.38 | Find a fundamental set of solutions: $\left[(D-1)^{4}-16\right] y=0$ | $\left\{e^{-x}, e^{3 x}, e^{x} \cos 2 x, e^{x} \sin 2 x\right\}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.2.14 | A 1200-gallon tank initially contains 40 pounds of salt dissolved in 600 gallons of water. Starting at $t_{0}=0$, water that contains $1 / 2$ pound of salt per gallon is added to the tank at the rate of 6 $\mathrm{gal} / \mathrm{min}$ and the resulting mixture is drained from the tank at $4 \mathrm{gal} / \mathrm{min}$. Find the quantity $Q(t)$ of salt in the tank at any time $t>0$ prior to overflow. | $Q=t+300-\frac{234 \times 10^{5}}{(t+300)^{2}}, \quad 0 \leq t \leq 300$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.5.3 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rr}-7 & 4 \\ -1 & -11\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{r}
-2 \\
1
\end{array}\right] e^{-9 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
0
\end{array}\right] e^{-9 t}+\left[\begin{array}{r}
-2 \\
1
\end{array}\right] t e^{-9 t}\right) | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.6.5 | Find the general solution: $y^{\prime \prime}-2 y^{\prime}+y=7 x^{3 / 2} e^{x} ; \quad y_{1}=e^{x}$ | $y=e^{x}\left(\frac{4}{5} x^{7 / 2}+c_{1}+c_{2} x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.4.2 | Find the general solution of the given Euler equation on $(0, \infty)$: $x^{2} y^{\prime \prime}-7 x y^{\prime}+7 y=0$ | $y=c_{1} x+c_{2} x^{7}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.5.1.16 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $4 x^{2} y^{\prime \prime}-4 x y^{\prime}+\left(3-16 x^{2}\right) y=0$, given that $y_{1}=x^{1 / 2} e^{2 x}$. | $y_{2}=x^{1 / 2} e^{-2 x}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.1.12 | Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $\left(1+3 x^{2}\right) y^{\prime \prime}+3 x^{2} y^{\prime}-2 y$. | $b_{0}=2 a_{2}-2 a_{0} b_{n}=(n+2)(n+1) a_{n+2}+[3 n(n-1)-2] a_{n}+3(n-1) a_{n-1}, n \geq 1$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.3.61 | Find the general solution: $y^{\prime \prime \prime}+y^{\prime \prime}-2 y=-e^{3 x}\left(9+67 x+17 x^{2}\right)$ | $y=e^{3 x}\left(1-x-\frac{x^{2}}{2}\right)+c_{1} e^{x}+e^{-x}\left(c_{2} \cos x+c_{3} \sin x\right)$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.4.5.25 | Find the orthogonal trajectories of the given family of curves: $x^{2}+2 y^{2}=c^{2}$ | $y=k x^{2}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |