
# 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

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

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

##### 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 $$-1<x<1,\: y(-1)=0,\: y(1)+y'(1)=0$$
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 $$y$$.

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

##### 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.

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

##### 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.

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