2.8: Problems
- Page ID
- 90923
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Solve the following initial value problems.
- \(x''+x=0,\quad x(0)=2,\quad x'(0)=0\).
- \(y''+2y'-8y=0,\quad y(0)=1,\quad y'(0)=2\).
- \(x^2y''-2xy'-4y=0,\quad y(1)=1,\quad y'(1)=0\).
Solve the following boundary value problems directly, when possible.
- \(x''+x=2,\quad x(0)=0,\quad x'(1)=0\).
- \(y''+2y'-8y=0,\quad y(0)=1,\quad y(1)=0\).
- \(y''+y=0,\quad y(0)=1,\quad y(\pi )=0\).
Consider the boundary value problem for the deflection of a horizontal beam fixed at one end,
\[\frac{d^4y}{dx^4}=C,\quad y(0)=0,\quad y'(0)=0,\quad y''(L)=0,\quad y'''(L)=0.\nonumber \]
Solve this problem assuming that \(C\) is a constant.
Find the product solutions, \(u(x, t) = T(t)X(x)\), to the heat equation, \(u_t − u_{xx} = 0\), on \([0, π]\) satisfying the boundary conditions \(u_x(0, t) = 0\) and \(u(π, t) = 0\).
Find the product solutions, \(u(x, t) = T(t)X(x)\), to the wave equation \(u_{tt} = 2u_{xx}\), on \([0, 2π]\) satisfying the boundary conditions \(u(0, t) = 0\) and \(u_x(2π, t) = 0\).
Find product solutions, \(u(x, t) = X(x)Y(y)\), to Laplace’s equation, \(u_{xx} + u_{yy} = 0\), on the unit square satisfying the boundary conditions \(u(0, y) = 0\), \(u(1, y) = g(y)\), \(u(x, 0) = 0\), and \(u(x, 1) = 0\).
Consider the following boundary value problems. Determine the eigenvalues, \(λ\), and eigenfunctions, \(y(x)\) for each problem.
- \(y'' + λy = 0,\quad y(0) = 0,\quad y'(1) = 0\).
- \(y'' − λy = 0,\quad y(−π) = 0,\quad y'(π) = 0\).
- \(x^2y'' + xy' + λy = 0,\quad y(1) = 0,\quad y(2) = 0\).
- \((x^2y')'+\lambda y=0,\quad y(1)=0,\quad y'(e)=0\).
In problem d you will not get exact eigenvalues. Show that you obtain a transcendental equation for the eigenvalues in the form \(\tan z = 2z\). Find the first three eigenvalues numerically.
Classify the following equations as either hyperbolic, parabolic, or elliptic.
- \(u_{yy} + u_{xy} + u_{xx} = 0\).
- \(3u_{xx} + 2u_{xy} + 5u_{yy} = 0\).
- \(x^2u_{xx}+2xyu_{xy}+y^2u_{yy}=0\).
- \(y^2u_{xx}+2xyu_{xy}+(x^2+4x^4)u_{yy}=0\).
Use d’Alembert’s solution to prove
\[f(−ζ) = f(ζ),\quad g(−ζ) = g(ζ)\nonumber \]
for the semi-infinite string satisfying the free end condition \(u_x(0, t) = 0\).
Derive a solution similar to d’Alembert’s solution for the equation \(u_{tt} + 2u_{xt} − 3u = 0\).
Construct the appropriate periodic extension of the plucked string initial profile given by
\[f(x)=\left\{\begin{array}{cc}x,&0\leq x\leq\frac{\ell}{2}, \\ \ell -x,&\frac{\ell}{2}\leq x\leq\ell ,\end{array}\right.\nonumber \]
satisfying the boundary conditions at \(u(0, t) = 0\) and \(u_x(\ell, t) = 0\) for \(t > 0\).
Find and sketch the solution of the problem
\[\begin{aligned} u_{tt}&=u_{xx},\quad 0\leq x\leq 1,\quad t> o \\ u(x,0)&=\left\{\begin{array}{rr}0,&0\leq x<\frac{1}{4}, \\ 1,&\frac{1}{4}\leq x\leq\frac{3}{4}, \\ 0,&\frac{3}{4}<x\leq 1,\end{array}\right. \\ u_t(x,0)&=0, \\ u(0,t)&=0,\quad t>0, \\ u(1,t)&=0,\quad t>0,\end{aligned} \nonumber \]