5.E: Eigenvalue Problems (Exercises)

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

Exercise $\PageIndex{5.1.1}$

Find eigenvalues and eigenfunctions of

$$y''+ \lambda y=0,~~~y(0)-y'(0)=0,~~~y(1)=0.$$

Exercise $\PageIndex{5.1.2}$

Expand the function $f(x)=x$ on $0 \leq x \leq 1$ using the eigenfunctions of the system

$$y''+ \lambda y=0,~~~y'(0)=0,~~~y(1)=0.$$

Exercise $\PageIndex{5.1.3}$

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.

Exercise $\PageIndex{5.1.4}$

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.

Exercise $\PageIndex{5.1.5}$: (more challenging)

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 $\lambda \geq 0$.

Exercise $\PageIndex{5.1.6}$

Find eigenvalues and eigenfunctions of

$$y''+ \lambda y=0,~~~y(-1)=0,~~~y(1)=0.$$

Answer

$\lambda_{n}=\frac{(2n-1)\pi}{2},\:n=1,\: 2,\: 3,\cdots ,$ $y_{n}=\cos\left(\frac{(2n-1)\pi}{2}x\right)$

Exercise $\PageIndex{5.1.7}$

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.

1. $xy''+\lambda y=0$ for $0<x<1,\: y(0)=0,\: y(1)=0,$
2. $(1+x^2)y''+2xy'+(\lambda -x^2)y=0$ for $-1<x<1,\: y(-1)=0,\: y(1)+y'(1)=0$

Answer

1. $p(x)=1,\: q(x)=0,\: r(x)=\frac{1}{x},\:\alpha_{1}=1,\:\alpha_{2}=0,\:\beta_{1}=1,\:\beta_{2}=0$. The problem is not regular.
2. $p(x)=1+x^{2},\: q(x)=x^{2},\: r(x)=1,\:\alpha_{1}=1,\:\alpha_{2}=0,\:\beta_{1}=1,\:\beta_{2}=1$. The problem is regular.

5.2: Application of eigenfunction series

Exercise $\PageIndex{5.2.1}$

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.

Exercise $\PageIndex{5.2.2}$

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.

Exercise $\PageIndex{5.2.3}$

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?

Exercise $\PageIndex{5.2.4}$

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.

Exercise $\PageIndex{5.2.5}$

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

$y(x,t)=\sin (\pi x)\cos (2\pi^{2}t)$

Exercise $\PageIndex{5.2.6}$

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.

Answer

$9y_{xxxx}+y_{tt}=0$ $(0<x<10,\: t>0)$, $\quad y(0,t)=y_{x}(0,t)=0$, $\quad y(10,t)=y_{x}(10,t)=0$, $\quad y(x,0)=\sin (\pi x)$, $\quad y_{t}(x,0)=x(10-x)$.

5.3: Steady periodic solutions

Exercise $\PageIndex{5.3.1}$

Suppose that the forcing function for the vibrating string is $F_0 \sin(\omega t)$. Derive the particular solution $y_p$.

Exercise $\PageIndex{5.3.2}$

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 $\PageIndex{5.3.1}$.

Exercise $\PageIndex{5.3.3}$

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.

Exercise $\PageIndex{5.3.4}$

Derive the solution for underground temperature oscillation without assuming that $T_0=0$.

Exercise $\PageIndex{5.3.5}$

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.

Answer

$y_{p}(x,t)=\sum\limits_{\overset{n=1}{n\text{ odd}}}^\infty \frac{-4}{n^{4}\pi^{4}}\left(\cos (n\pi x)-\frac{\cos (n\pi )-1}{\sin (n\pi )}\sin (n\pi x)-1\right)\cos (n\pi t).$

Exercise $\PageIndex{5.3.6}$

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.

Answer

Approximately 1991 centimeters