5.E: Eigenvalue Problems (Exercises)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
5.1: Sturm-Liouville problems
Find eigenvalues and eigenfunctions of
\[y''+ \lambda y=0,~~~y(0)-y'(0)=0,~~~y(1)=0.\]
Expand the function \(f(x)=x\) on \(0 \leq x \leq 1\) using the eigenfunctions of the system
\[y''+ \lambda y=0,~~~y'(0)=0,~~~y(1)=0.\]
Suppose that you had a Sturm-Liouville problem on the interval \([0,1]\) and came up with \(y_n(x)=\sin(\gamma nx)\), where \(\gamma >0\) is some constant. Decompose \(f(x)=x, 0<x<1\), in terms of these eigenfunctions.
Find eigenvalues and eigenfunctions of
\[y'^{(4)}+ \lambda y=0,~~~y(0)=0,~~~y'(0)=0,~~~y(1)=0~~~y'(1)=0.\]
This problem is not a Sturm-Liouville problem, but the idea is the same.
Find eigenvalues and eigenfunctions for
\[\frac{d}{dx}(e^xy')+ \lambda e^xy=0,~~~y(0)=0,~~~y(1)=0.\]
Hint: First write the system as a constant coefficient system to find general solutions. Do note that Theorem 5.1.1 guarantees \(\lambda \geq 0\).
Find eigenvalues and eigenfunctions of
\[y''+ \lambda y=0,~~~y(-1)=0,~~~y(1)=0.\]
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\(\lambda_{n}=\frac{(2n-1)\pi}{2},\:n=1,\: 2,\: 3,\cdots ,\) \(y_{n}=\cos\left(\frac{(2n-1)\pi}{2}x\right)\)
Put the following problems into the standard form for Sturm-Liouville problems, that is, find \(p(x),q(x), r(x), \alpha_1,\alpha_,\beta_1,\beta_1, \), and decide if the problems are regular or not.
- \(xy''+\lambda y=0\) for \(0<x<1,\: y(0)=0,\: y(1)=0,\)
- \((1+x^2)y''+2xy'+(\lambda -x^2)y=0\) for \(-1<x<1,\: y(-1)=0,\: y(1)+y'(1)=0\)
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- \(p(x)=1,\: q(x)=0,\: r(x)=\frac{1}{x},\:\alpha_{1}=1,\:\alpha_{2}=0,\:\beta_{1}=1,\:\beta_{2}=0\). The problem is not regular.
- \(p(x)=1+x^{2},\: q(x)=x^{2},\: r(x)=1,\:\alpha_{1}=1,\:\alpha_{2}=0,\:\beta_{1}=1,\:\beta_{2}=1\). The problem is regular.
5.2: Application of eigenfunction series
Suppose you have a beam of length \(5\) with free ends. Let \(y\) be the transverse deviation of the beam at position \(x\) on the beam \((0<x<5)\). You know that the constants are such that this satisfies the equation \(y_{tt}+4y_{xxxx}=0\). Suppose you know that the initial shape of the beam is the graph of \(x(5-x)\), and the initial velocity is uniformly equal to \(2\) (same for each \(x\)) in the positive \(y\) direction. Set up the equation together with the boundary and initial conditions. Just set up, do not solve.
Suppose you have a beam of length \(5\) with one end free and one end fixed (the fixed end is at \(x=5\)). Let \(u\) be the longitudinal deviation of the beam at position \(x\) on the beam \((0<x<5)\). You know that the constants are such that this satisfies the equation \(u_{tt}=4u_{xx}\). Suppose you know that the initial displacement of the beam is \(\frac{x-5}{50}\), and the initial velocity is \(\frac{-(x-5)}{100}\) in the positive \(u\) direction. Set up the equation together with the boundary and initial conditions. Just set up, do not solve.
Suppose the beam is \(L\) units long, everything else kept the same as in (5.2.2). What is the equation and the series solution?
Suppose you have
\[\begin{align}\begin{aligned} & a^4 y_{xxxx} + y_{tt} = 0 \quad (0 < x < 1, t > 0) , \\ & y(0,t) = y_{xx}(0,t) = 0,\\ & y(1,t) = y_{xx}(1,t) = 0 ,\\ & y(x,0) = f(x), \quad y_{t}(x,0) = g(x) . \end{aligned}\end{align}\]
That is, you have also an initial velocity. Find a series solution. Hint: Use the same idea as we did for the wave equation.
Suppose you have a beam of length \(1\) with hinged ends. Let \(y\) be the transverse deviation of the beam at position \(x\) on the beam (\(0<x<1\)). You know that the constants are such that this satisfies the equation \(y_{tt}+4y_{xxxx}=0\). Suppose you know that the initial shape of the beam is the graph of \(\sin(\pi x)\), and the initial velocity is \(0\). Solve for \(y\).
- Answer
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\(y(x,t)=\sin (\pi x)\cos (2\pi^{2}t)\)
Suppose you have a beam of length \(10\) with two fixed ends. Let \(y\) be the transverse deviation of the beam at position \(x\) on the beam (\(0<x<10\)). You know that the constants are such that this satisfies the equation \(y_{tt}+9y_{xxxx}=0\). Suppose you know that the initial shape of the beam is the graph of \(\sin(\pi x)\), and the initial velocity is uniformly equal to \(x(10-x)\). Set up the equation together with the boundary and initial conditions. Just set up, do not solve.
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\(9y_{xxxx}+y_{tt}=0\) \((0<x<10,\: t>0)\), \(\quad y(0,t)=y_{x}(0,t)=0\), \(\quad y(10,t)=y_{x}(10,t)=0\), \(\quad y(x,0)=\sin (\pi x)\), \(\quad y_{t}(x,0)=x(10-x)\).
5.3: Steady periodic solutions
Suppose that the forcing function for the vibrating string is \(F_0 \sin(\omega t)\). Derive the particular solution \(y_p\).
Take the forced vibrating string. Suppose that \(L=1,a=1\). Suppose that the forcing function is the square wave that is \(1\) on the interval \(0<x<1\) and \(-1 \)on the interval \(-1<x<0\). Find the particular solution. Hint: You may want to use result of Exercise \(\PageIndex{5.3.1}\).
The units are cgs (centimeters-grams-seconds). For \(k=0.005, \omega =1.991 \times 10^{-7},A_0=20\). Find the depth at which the temperature variation is half (\(\pm 10\) degrees) of what it is on the surface.
Derive the solution for underground temperature oscillation without assuming that \(T_0=0\).
Take the forced vibrating string. Suppose that \(L=1,a=1\). Suppose that the forcing function is a sawtooth, that is \(|x|-\frac{1}{2}\) on \(-1<x<1\) extended periodically. Find the particular solution.
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\(y_{p}(x,t)=\sum\limits_{\overset{n=1}{n\text{ odd}}}^\infty \frac{-4}{n^{4}\pi^{4}}\left(\cos (n\pi x)-\frac{\cos (n\pi )-1}{\sin (n\pi )}\sin (n\pi x)-1\right)\cos (n\pi t).\)
The units are cgs (centimeters-grams-seconds). For \(k=0.01, \omega =1.991 \times 10^{-7},A_0=25\). Find the depth at which the summer is again the hottest point.
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Approximately 1991 centimeters