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college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.9.2.12 | Find the general solution: $6 y^{(4)}+5 y^{\prime \prime \prime}+7 y^{\prime \prime}+5 y^{\prime}+y=0$ | $y=c_{1} e^{-x / 2}+c_{2} e^{-x / 3}+c_{3} \cos x+c_{4} \sin x$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.10.5.24 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}5 & -1 & 1 \\ -1 & 9 & -3 \\ -2 & 2 & 4\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] e^{6 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] \frac{e^{6 t}}{4}+\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] t e^{6 t}\right)+c_{3}\left(\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \frac{e^{6 t}}{8}+\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] \frac{t e^{6 t}}{4}+\left[\begin{array}{l}
0 \\
1 \\
1
\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 |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.7.6.3 | Find the general solution: $x y^{\prime}+(\ln x) y=0$ | $y_{1}=1+x-x^{2}+\frac{1}{3} x^{3}+\cdots$
$y_{2}=y_{1} \ln x-x\left(3-\frac{1}{2} x-\frac{31}{18} x^{2}+\cdots\right)$ | 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.5.1.23 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $\left(x^{2}-2 x\right) y^{\prime \prime}+\left(2-x^{2}\right) y^{\prime}+(2 x-2) y=0$, given that $y_{1}=e^{x}$. | $y_{2}=x^{2}$ | 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.5.1.15 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $x^{2} y^{\prime \prime}-(2 a-1) x y^{\prime}+a^{2} y=0$ $(a=$ nonzero constant $)$, $x>0$, given that $y_{1}=x^{a}$. | $y_{2}=x^{a} \ln x$ | 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.2.2.25 | Solve the equation $y^{\prime} \sqrt{1-x^{2}}+\sqrt{1-y^{2}}=0$ explicitly. Hint: Use the identity $\sin (A-B)=\sin A \cos B-$ $\cos A \sin B$. | $y=-x \cos c+\sqrt{1-x^{2}} \sin c ; \quad y \equiv 1 ; y \equiv-1$ | 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.10.5.26 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}-6 & -4 & -4 \\ 2 & -1 & 1 \\ 2 & 3 & 1\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] e^{-2 t}+c_{2}\left(\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] e^{-2 t}+\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] t e^{-2 t}\right)+c_{3}\left(\left[\begin{array}{r}
3 \\
-2 \\
0
\end{array}\right] \frac{e^{-2 t}}{4}+\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] t e^{-2 t}+\left[\begin{array}{r}
0 \\
-1 \\
1
\end{array}\right] \frac{t^{2} e^{-2 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.10.5.27 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}0 & 2 & -2 \\ -1 & 5 & -3 \\ 1 & 1 & 1\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] e^{2 t}+c_{2}\left(\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \frac{e^{2 t}}{2}+\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] t e^{2 t}\right)+c_{3}\left(\left[\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right] \frac{e^{2 t}}{8}+\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] \frac{t e^{2 t}}{2}+\left[\begin{array}{l}
0 \\
1 \\
1
\end{array}\right] \frac{t^{2} e^{2 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.7.1.14 | Find a power series solution $y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ for $\left(1+x^{2}\right) y^{\prime \prime}+(2-x) y^{\prime}+3 y$. | $b_{n}=(n+2)(n+1) a_{n+2}+2(n+1) a_{n+1}+\left(n^{2}-2 n+3\right) a_{n}$ | 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.2.2.3 | Find all solutions: $\left(3 y^{3}+3 y \cos y+1\right) y^{\prime}+\frac{(2 x+1) y}{1+x^{2}}=0$ | $y=\frac{c}{x-c} \quad y \equiv-1$ | 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.9.2.4 | Find the general solution: $2 y^{\prime \prime \prime}+3 y^{\prime \prime}-2 y^{\prime}-3 y=0$ | $y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} e^{-3 x / 2}$ | 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.2.2.37 | Solve the equation using variation of parameters followed by separation of variables: $y^{\prime}-y=\frac{(x+1) e^{4 x}}{\left(y+e^{x}\right)^{2}}$ | $y=e^{x}\left(-1+\left(3 x e^{x}+c\right)^{1 / 3}\right)$ | 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.4.2.1 | A thermometer is moved from a room where the temperature is $70^{\circ} \mathrm{F}$ to a freezer where the temperature is $12^{\circ} \mathrm{F}$. After 30 seconds the thermometer reads $40^{\circ} \mathrm{F}$. What does it read after 2 minutes? | $\approx 15.15^{\circ} \mathrm{F}$ | 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.5.1.18 | Find the Wronskian of a given set $\left\{y_{1}, y_{2}\right\}$ of solutions of $x^{2} y^{\prime \prime}-2 x y^{\prime}+\left(x^{2}+2\right) y=0$, given that $y_{1}=x \cos x$. | $y_{2}=x \sin x$ | 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.5.2.6 | Find the general solution: $y^{\prime \prime}+6 y^{\prime}+10 y=0$ | $y=e^{-3 x}\left(c_{1} \cos x+c_{2} \sin x\right)$ | 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.5.6.3 | Find the general solution: $x^{2} y^{\prime \prime}-x y^{\prime}+y=x ; \quad y_{1}=x$ | $y=\frac{x(\ln |x|)^{2}}{2}+c_{1} x+c_{2} x \ln |x|$ | 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.10.5.12 | Find the general solution: $\mathbf{y}^{\prime}=\left[\begin{array}{rrr}6 & -5 & 3 \\ 2 & -1 & 3 \\ 2 & 1 & 1\end{array}\right] \mathbf{y}$ | \mathbf{y}=c_{1}\left[\begin{array}{r}
-1 \\
-1 \\
1
\end{array}\right] e^{-2 t}+c_{2}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] e^{4 t}+c_{3}\left(\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right] \frac{e^{4 t}}{2}+\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] t e^{4 t}\right) | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |
college_math.ELEMENTARY_DIFFERENTIAL_EQUATIONS | exercise.2.4.4 | Solve the given Bernoulli equation: $\left(1+x^{2}\right) y^{\prime}+2 x y=\frac{1}{\left(1+x^{2}\right) y}$ | $y= \pm \frac{\sqrt{2 x+c}}{1+x^{2}}$ | Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (CC BY-NC-SA 3.0) | college_math.differential_equation |