5.E: Eigenvalue Problems (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
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″+λy=0, y(0)−y′(0)=0, y(1)=0.
Expand the function f(x)=x on 0≤x≤1 using the eigenfunctions of the system
y″+λy=0, y′(0)=0, y(1)=0.
Suppose that you had a Sturm-Liouville problem on the interval [0,1] and came up with yn(x)=sin(γnx), where γ>0 is some constant. Decompose f(x)=x,0<x<1, in terms of these eigenfunctions.
Find eigenvalues and eigenfunctions of
y′(4)+λy=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
ddx(exy′)+λexy=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 λ≥0.
Find eigenvalues and eigenfunctions of
y″+λy=0, y(−1)=0, y(1)=0.
- Answer
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λn=(2n−1)π2,n=1,2,3,⋯, yn=cos((2n−1)π2x)
Put the following problems into the standard form for Sturm-Liouville problems, that is, find p(x),q(x),r(x),α1,α,β1,β1,, and decide if the problems are regular or not.
- xy″+λy=0 for 0<x<1,y(0)=0,y(1)=0,
- (1+x2)y″+2xy′+(λ−x2)y=0 for −1<x<1,y(−1)=0,y(1)+y′(1)=0
- Answer
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- p(x)=1,q(x)=0,r(x)=1x,α1=1,α2=0,β1=1,β2=0. The problem is not regular.
- p(x)=1+x2,q(x)=x2,r(x)=1,α1=1,α2=0,β1=1,β2=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 ytt+4yxxxx=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 utt=4uxx. Suppose you know that the initial displacement of the beam is x−550, and the initial velocity is −(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
a4yxxxx+ytt=0(0<x<1,t>0),y(0,t)=yxx(0,t)=0,y(1,t)=yxx(1,t)=0,y(x,0)=f(x),yt(x,0)=g(x).
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 ytt+4yxxxx=0. Suppose you know that the initial shape of the beam is the graph of sin(πx), and the initial velocity is 0. Solve for y.
- Answer
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y(x,t)=sin(πx)cos(2π2t)
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 ytt+9yxxxx=0. Suppose you know that the initial shape of the beam is the graph of sin(πx), 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
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9yxxxx+ytt=0 (0<x<10,t>0), y(0,t)=yx(0,t)=0, y(10,t)=yx(10,t)=0, y(x,0)=sin(πx), yt(x,0)=x(10−x).
5.3: Steady periodic solutions
Suppose that the forcing function for the vibrating string is F0sin(ωt). Derive the particular solution yp.
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 −1on the interval −1<x<0. Find the particular solution. Hint: You may want to use result of Exercise 5.E.5.3.1.
The units are cgs (centimeters-grams-seconds). For k=0.005,ω=1.991×10−7,A0=20. Find the depth at which the temperature variation is half (±10 degrees) of what it is on the surface.
Derive the solution for underground temperature oscillation without assuming that T0=0.
Take the forced vibrating string. Suppose that L=1,a=1. Suppose that the forcing function is a sawtooth, that is |x|−12 on −1<x<1 extended periodically. Find the particular solution.
- Answer
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yp(x,t)=∞∑n=1n odd−4n4π4(cos(nπx)−cos(nπ)−1sin(nπ)sin(nπx)−1)cos(nπt).
The units are cgs (centimeters-grams-seconds). For k=0.01,ω=1.991×10−7,A0=25. Find the depth at which the summer is again the hottest point.
- Answer
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Approximately 1991 centimeters