Content check — Differential Equations (Lebl), Chapter 6: More on eigenvalue problems

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Chapter 6: More on eigenvalue problems

6 problems · 0 figures

Exercise 5.1.101 §5.1 source ↗

Find eigenvalues and eigenfunctions of

$y^{''} + λ y = 0, y (- 1) = 0, y (1) = 0.$

Answer

$λ_{n} = \frac{n^{2} π^{2}}{4}$ ⁠, $n = 1,2,3, \dots$ ⁠. For odd $n$ ⁠, $y_{n} = \cos (\frac{n π}{2} x)$ ⁠, for even $n$ ⁠, $y_{n} = \sin (\frac{n π}{2} x)$ ⁠.

Exercise 5.1.102 §5.1 source ↗

Put the following problems into the standard form for Sturm–Liouville problems, that is, find $p (x)$ ⁠, $q (x)$ ⁠, $r (x)$ ⁠, $α_{1}$ ⁠, $α_{2}$ ⁠, $β_{1}$ ⁠, and $β_{2}$ ⁠, and decide if the problems are regular or not.

a) $x y^{''} + λ y = 0$ for $0 < x < 1$ ⁠, $y (0) = 0$ ⁠, $y (1) = 0$ ⁠.

b) $(1 + x^{2}) y^{''} +$ $2 x y^{'} +$ $(λ - x^{2}) y =$ $0$ for $- 1 < x < 1$ ⁠, $y (- 1) = 0$ ⁠, $y (1) + y^{'} (1) = 0$ ⁠. (In an earlier version of this book, a typo rendered the equation as $(1 + x^{2}) y^{''} -$ $2 x y^{'} +$ $(λ - x^{2}) y =$ $0$ ending up with something harder than intended. Try this equation for a further challenge.)

Answer

a) $p (x) = 1$ ⁠, $q (x) = 0$ ⁠, $r (x) = \frac{1}{x}$ ⁠, $α_{1} = 1$ ⁠, $α_{2} = 0$ ⁠, $β_{1} = 1$ ⁠, $β_{2} = 0$ ⁠. The problem is not regular. b) $p (x) = 1 + x^{2}$ ⁠, $q (x) = x^{2}$ ⁠, $r (x) = 1$ ⁠, $α_{1} = 1$ ⁠, $α_{2} = 0$ ⁠, $β_{1} = 1$ ⁠, $β_{2} = 1$ ⁠. The problem is regular.

Exercise 5.2.101 §5.2 source ↗

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 $y_{t t} + 4 y_{x x x x} = 0$ ⁠. Suppose you know that the initial shape of the beam is the graph of $\sin (π x)$ ⁠, and the initial velocity is 0. Solve for $y$ ⁠.

Answer

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

Exercise 5.2.102 §5.2 source ↗

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 $y_{t t} + 9 y_{x x x x} = 0$ ⁠. Suppose you know that the initial shape of the beam is the graph of $\sin^{2} (π x)$ ⁠, and the initial velocity is equal to $x (10 - x)$ ⁠. Set up the equation together with the boundary and initial conditions. Just set up, do not solve.

Answer

$9 y_{x x x x} +$ $y_{t t} =$ $0$ $(0 < x < 10, t > 0)$ ⁠, $y (0, t) = y_{x} (0, t) = 0$ ⁠, $y (10, t) = y_{x} (10, t) = 0$ ⁠, $y (x, 0) =$ $\sin^{2} (π x),$ $y_{t} (x, 0) =$ $x (10 - x)$ ⁠.

Exercise 5.3.101 §5.3 source ↗

Take the forced vibrating string. Suppose that $L = 1$ ⁠, $a = 1$ ⁠. Suppose that the forcing function is a triangle wave, $| t | - \frac{π}{2}$ on $- π < t < π$ extended periodically. Find the particular solution.

Answer

$y_{p} (x, t) =$ $\sum_{\begin{matrix} n = 1 \\ n odd \end{matrix}}^{\infty} \frac{- 4}{π n^{4}} (\cos (n x) -$ $\frac{\cos (n) - 1}{\sin (n)} \sin (n x) -$ $1) \cos (n t) .$

Exercise 5.3.102 §5.3 source ↗

The units are cgs (centimeters-grams-seconds). For $k = 0.01$ ⁠, $ω = 1.991 \times 10^{- 7}$ ⁠, $A_{0} = 25$ ⁠. Find the depth at which the summer is again the hottest point.

Answer

Approximately 1991 centimeters

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Exercises and answers are adapted from Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl, which is free to read and download, and are used under the CC BY-NC-SA 4.0 licence. They are taken from the book’s LaTeX source and set for the web, so they may differ slightly from the printed page. This adaptation is released under the same licence and is neither affiliated with nor endorsed by Lebl or Jiří Lebl.