17.4E: Ejercicios para la Sección 17.4
- Page ID
- 116308
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Encuentre una solución de series de potencia para cada una de las siguientes ecuaciones diferenciales.
1. \(y″+6y′=0\)
2. \(5y″+y′=0\)
- Contestar
- \(\displaystyle y=c_0+5c_1 \sum_{n=1}^∞ \frac{(−x/5)^n}{n!}=c_0+5c_1e^{−x/5}\)
3. \(y''+25y=0\)
4. \(y''−y=0\)
- Contestar
- \(\displaystyle y=c_0 \sum_{n=0}^∞ \frac{(x)^{2n}}{(2n)!}+c_1 \sum_{n=0}^∞ \frac{(x)^{2n+1}}{(2n+1)!}\)
5. \(2y′+y=0\)
6. \(y′−2xy=0\)
- Contestar
- \(\displaystyle y=c_0 \sum_{n=0}^∞ \frac{x^{2n}}{n!}=c_0e^{x^2}\)
7. \((x−7)y′+2y=0\)
8. \(y''−xy′−y=0\)
- Contestar
- \(\displaystyle y=c_0 \sum_{n=0}^∞ \frac{x^{2n}}{2^nn!}+c_1 \sum_{n=0}^∞ \frac{x^{2n+1}}{1⋅3⋅5⋅7⋯(2n+1)}\)
9. \((1+x^2)y''−4xy′+6y=0\)
10. \(x^2y''−xy′−3y=0\)
- Contestar
- \(y=c_1x^3+\dfrac{c_2}{x}\)
11. \(y″−8y′=0, \quad y(0)=−2, \; y′(0)=10\)
12. \(y″−2xy=0, \quad y(0)=1, \; y′(0)=−3\)
- Contestar
- \(y=1−3x+\dfrac{2x^3}{3!}−\dfrac{12x^4}{4!}+\dfrac{16x^6}{6!}−\dfrac{120x^7}{7!}+⋯\)
13. La ecuación diferencial\(x^2y″+xy′+(x^2−1)y=0\) es una ecuación de orden de Bessel\(1.\) Utilice una serie de potencias de la forma\(\displaystyle y=\sum_{n=0}^∞ a_nx^n\) para encontrar la solución.