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In Chapter 12 we’ll study partial differential equations that arise in problems of heat conduction, wave propagation, and potential theory. The purpose of this chapter is to develop tools required to solve these equations. In this section we consider the following problems, where \(\lambda\) is a real number and \(L>0\):
- Problem 1: \(y''+\lambda y=0,\quad y(0)=0,\quad y(L)=0\)
- Problem 2: \(y''+\lambda y=0,\quad y'(0)=0,\quad y'(L)=0\)
- Problem 3: \(y''+\lambda y=0,\quad y(0)=0,\quad y'(L)=0\)
- Problem 4: \(y''+\lambda y=0,\quad y'(0)=0,\quad y(L)=0\)
- Problem 5: \(y''+\lambda y=0,\quad y(-L)=y(L), \quad y'(-L)=y'(L)\)
In each problem the conditions following the differential equation are called boundary conditions. Note that the boundary conditions in Problem 5, unlike those in Problems 1-4, don’t require that \(y\) or \(y'\) be zero at the boundary points, but only that \(y\) have the same value at \(x=\pm L\), and that \(y'\) have the same value at \(x=\pm L\). We say that the boundary conditions in Problem 5 are periodic.
Obviously, \(y\equiv0\) (the trivial solution) is a solution of Problems 1-5 for any value of \(\lambda\). For most values of \(\lambda\), there are no other solutions. The interesting question is this:
For what values of \(\lambda\) does the problem have nontrivial solutions, and what are they?
A value of \(\lambda\) for which the problem has a nontrivial solution is an eigenvalue of the problem, and the nontrivial solutions are \(\lambda\)-eigenfunctions, or eigenfunctions associated with \(\lambda\). Note that a nonzero constant multiple of a \(\lambda\)-eigenfunction is again a \(\lambda\)-eigenfunction.
Problems 1-5 are called eigenvalue problems. Solving an eigenvalue problem means finding all its eigenvalues and associated eigenfunctions. We’ll take it as given here that all the eigenvalues of Problems 1-5 are real numbers. This is proved in a more general setting in Section 13.2.
Theorem 11.1.1
Problems \(1\)–\(5\) have no negative eigenvalues. Moreover\(,\) \(\lambda=0\) is an eigenvalue of Problems \(2\) and \(5,\) with associated eigenfunction \(y_0=1,\) but \(\lambda=0\) isn’t an eigenvalue of Problems \(1,\) \(3,\) or \(4\).
- Proof
-
We consider Problems 1-4, and leave Problem 5 to you (Exercise 11.1.1). If \(y''+\lambda y=0\), then \(y(y''+\lambda y)=0\), so
\[\int_0^L y(x)(y''(x)+\lambda y(x))\,dx=0;\nonumber \]
therefore,
\[\label{eq:11.1.1} \lambda\int_0^L y^2(x)\,dx=-\int_0^L y(x)y''(x)\, dx.\]
Integration by parts yields
\[\label{eq:11.1.2} \begin{array}{rcl} \int_0^L y(x)y''(x)\, dx &= \left.\ y(x)y'(x)\right| _{0}^{L} -\int_0^L (y'(x))^2\,dx \\ &=\ y(L)y'(L)-y(0)y'(0)-\int_0^L (y'(x))^2\,dx. \end{array}\]
However, if \(y\) satisfies any of the boundary conditions of Problems 1-4, then
\[y(L)y'(L)-y(0)y'(0)=0;\nonumber \]
hence, Equation \ref{eq:11.1.1} and Equation \ref{eq:11.1.2} imply that
\[\lambda\int_0^L y^2(x)\,dx=\int_0^L (y'(x))^2\, dx.\nonumber \]
If \(y\not\equiv0\), then \(\int_0^L y^2(x)\,dx>0\). Therefore \(\lambda\ge0\) and, if \(\lambda=0\), then \(y'(x)=0\) for all \(x\) in \((0,L)\) (why?), and \(y\) is constant on \((0,L)\). Any constant function satisfies the boundary conditions of Problem 2, so \(\lambda=0\) is an eigenvalue of Problem 2 and any nonzero constant function is an associated eigenfunction. However, the only constant function that satisfies the boundary conditions of Problems \(1\), \(3\), or \(4\) is \(y\equiv0\). Therefore \(\lambda=0\) isn’t an eigenvalue of any of these problems.
Example 11.1.1
Solve the eigenvalue problem
\[\label{eq:11.1.3} y''+\lambda y=0,\quad y(0)=0,\quad y(L)=0.\]
Solution
From Theorem 11.1.1, any eigenvalues of Equation \ref{eq:11.1.3} must be positive. If \(y\) satisfies Equation \ref{eq:11.1.3} with \(\lambda>0\), then
\[y=c_1\cos\sqrt\lambda\, x+c_2\sin\sqrt\lambda\, x,\nonumber \]
where \(c_1\) and \(c_2\) are constants. The boundary condition \(y(0)=0\) implies that \(c_1=0\). Therefore \(y=c_2\sin\sqrt\lambda\, x\). Now the boundary condition \(y(L)=0\) implies that \(c_2\sin\sqrt\lambda\, L=0\). To make \(c_2\sin\sqrt\lambda\, L=0\) with \(c_2\ne0\), we must choose \(\sqrt\lambda=n\pi/L\), where \(n\) is a positive integer. Therefore \(\lambda_n=n^2\pi^2/L^2\) is an eigenvalue and
\[y_n=\sin{n\pi x\over L}\nonumber \]
is an associated eigenfunction.
For future reference, we state the result of Example 11.1.1 as a theorem.
Theorem 11.1.2
The eigenvalue problem
\[y''+\lambda y=0,\quad y(0)=0,\quad y(L)=0\nonumber \]
has infinitely many positive eigenvalues \(\lambda_n=n^2\pi^2/L^2\), with associated eigenfunctions
\[y_n=\sin {n\pi x\over L},\quad n=1,2,3,\dots.\nonumber\]
There are no other eigenvalues.
We leave it to you to prove the next theorem about Problem 2 by an argument like that of Example 11.1.1 (Exercise 11.1.17).
Theorem 11.1.3
The eigenvalue problem
\[y''+\lambda y=0,\quad y'(0)=0,\quad y'(L)=0\nonumber \]
has the eigenvalue \(\lambda_0=0\), with associated eigenfunction \(y_0=1\), and infinitely many positive eigenvalues \(\lambda_n=n^2\pi^2/L^2\), with associated eigenfunctions
\[y_n=\cos {n\pi x\over L}, n=1,2,3\dots.\nonumber\]
There are no other eigenvalues.
Example 11.1.2
Solve the eigenvalue problem
\[\label{eq:11.1.4} y''+\lambda y=0,\quad y(0)=0,\quad y'(L)=0.\]
Solution:
From Theorem 11.1.1, any eigenvalues of Equation \ref{eq:11.1.4} must be positive. If \(y\) satisfies Equation \ref{eq:11.1.4} with \(\lambda>0\), then
\[y=c_1\cos\sqrt\lambda\, x+c_2\sin\sqrt\lambda\, x,\nonumber \]
where \(c_1\) and \(c_2\) are constants. The boundary condition \(y(0)=0\) implies that \(c_1=0\). Therefore \(y=c_2\sin\sqrt\lambda\, x\). Hence, \(y'=c_2\sqrt\lambda\cos\sqrt\lambda\,x\) and the boundary condition \(y'(L)=0\) implies that \(c_2\cos\sqrt\lambda\,L=0\). To make \(c_2\cos\sqrt\lambda\,L=0\) with \(c_2\ne0\) we must choose
\[\sqrt\lambda={(2n-1)\pi\over2L},\nonumber \]
where \(n\) is a positive integer. Then \(\lambda_n=(2n-1)^2\pi^2/4L^2\) is an eigenvalue and
\[y_n=\sin{(2n-1)\pi x\over2L}\nonumber \]
is an associated eigenfunction.
For future reference, we state the result of Example 11.1.2 as a theorem.
Theorem 11.1.4
The eigenvalue problem
\[y''+\lambda y=0,\quad y(0)=0,\quad y'(L)=0\nonumber \]
has infinitely many positive eigenvalues \(\lambda_n=(2n-1)^2\pi^2/4L^2,\) with associated eigenfunctions
\[y_n=\sin{(2n-1)\pi x\over2L},\quad n=1,2,3,\dots.\nonumber \]
There are no other eigenvalues.
We leave it to you to prove the next theorem about Problem 4 by an argument like that of Example 11.1.2 (Exercise 11.1.18).
Theorem 11.1.5
The eigenvalue problem
\[y''+\lambda y=0,\quad y'(0)=0,\quad y(L)=0\nonumber \]
has infinitely many positive eigenvalues \(\lambda_n=(2n-1)^2\pi^2/4L^2,\) with associated eigenfunctions
\[y_n=\cos{(2n-1)\pi x\over2L},\quad n=1,2,3,\dots.\nonumber\]
There are no other eigenvalues.
Example 11.1.3
Solve the eigenvalue problem
\[\label{eq:11.1.5} y''+\lambda y=0,\quad y(-L)=y(L),\quad y'(-L)=y'(L).\]
Solution
From Theorem 11.1.1, \(\lambda=0\) is an eigenvalue of Equation \ref{eq:11.1.5} with associated eigenfunction \(y_0=1\), and any other eigenvalues must be positive. If \(y\) satisfies Equation \ref{eq:11.1.5} with \(\lambda>0\), then
\[\label{eq:11.1.6} y=c_1\cos\sqrt\lambda\, x+c_2\sin\sqrt\lambda\, x,\]
where \(c_1\) and \(c_2\) are constants. The boundary condition \(y(-L)=y(L)\) implies that
\[\label{eq:11.1.7} c_1\cos(-\sqrt\lambda\,L)+c_2\sin(-\sqrt\lambda\,L)=c_1\cos \sqrt\lambda\,L+c_2\sin \sqrt\lambda\,L.\]
Since
\[\label{eq:11.1.8} \cos(-\sqrt\lambda\,L)=\cos \sqrt\lambda\,L\quad \text{and} \quad \sin(-\sqrt\lambda\,L)=-\sin \sqrt\lambda\,L,\]
Equation \ref{eq:11.1.7} implies that
\[\label{eq:11.1.9} c_2\sin \sqrt\lambda\,L=0.\]
Differentiating Equation \ref{eq:11.1.6} yields
\[y'=\sqrt\lambda\left(-c_1\sin\sqrt\lambda x+c_2\cos\sqrt\lambda x\right).\nonumber\]
The boundary condition \(y'(-L)=y'(L)\) implies that
\[-c_1\sin(-\sqrt\lambda\,L)+c_2\cos(-\sqrt\lambda\,L)=-c_1\sin \sqrt\lambda\,L+c_2\cos \sqrt\lambda\,L,\nonumber\]
and Equation \ref{eq:11.1.8} implies that
\[\label{eq:11.1.10} c_1\sin \sqrt\lambda\,L=0.\]
Eqns. Equation \ref{eq:11.1.9} and Equation \ref{eq:11.1.10} imply that \(c_1=c_2=0\) unless \(\sqrt\lambda =n\pi /L\), where \(n\) is a positive integer. In this case Equation \ref{eq:11.1.9} and Equation \ref{eq:11.1.10} both hold for arbitrary \(c_1\) and \(c_2\). The eigenvalue determined in this way is \(\lambda_n=n^2\pi^2/L^2\), and each such eigenvalue has the linearly independent associated eigenfunctions
\[\cos {n\pi x\over L} \quad \text{and} \quad \sin{ n\pi x\over L}. \nonumber\]
For future reference we state the result of Example 11.1.3 as a theorem.
Theorem 11.1.6
The eigenvalue problem
\[y''+\lambda y=0,\quad y(-L)=y(L),\quad y'(-L)=y'(L),\nonumber\]
has the eigenvalue \(\lambda_0=0\), with associated eigenfunction \(y_0=1\) and infinitely many positive eigenvalues \(\lambda_n=n^2\pi^2/L^2,\) with associated eigenfunctions
\[y_{1n}=\cos {n\pi x\over L} \quad \text{and} \quad y_{2n}=\sin {n\pi x\over L},\quad n=1,2,3,\dots.\nonumber\]
There are no other eigenvalues.
Orthogonality
We say that two integrable functions \(f\) and \(g\) are orthogonal on an interval \([a,b]\) if
\[\int_a^bf(x)g(x)\,dx=0.\nonumber\]
More generally, we say that the functions \(\phi_1\), \(\phi_2\), …, \(\phi_n\), …(finitely or infinitely many) are orthogonal on \([a,b]\) if
\[\int_a^b\phi_i(x)\phi_j(x)\,dx=0\quad \text{whenever} \quad i\ne j.\nonumber\]
The importance of orthogonality will become clear when we study Fourier series in the next two sections.
Example 11.1.4
Show that the eigenfunctions
\[\label{eq:11.1.11} 1,\, \cos{\pi x\over L},\, \sin{\pi x\over L}, \, \cos{2\pi x\over L}, \, \sin{2\pi x\over L},\dots, \cos{n\pi x\over L}, \, \sin{n\pi x\over L},\dots\]
of Problem 5 are orthogonal on \([-L,L]\).
Solution
We must show that
\[\label{eq:11.1.12} \int_{-L}^L f(x)g(x)\,dx=0\]
whenever \(f\) and \(g\) are distinct functions from Equation \ref{eq:11.1.11}. If \(r\) is any nonzero integer, then
\[\label{eq:11.1.13} \int_{-L}^L\cos{r\pi x\over L}\,dx ={L\over r\pi}\sin{r\pi x\over L}\bigg|_{-L}^L=0.\]
and
\[\int_{-L}^L\sin{r\pi x\over L}\,dx =-{L\over r\pi}\cos{r\pi x\over L}\bigg|_{-L}^L=0.\nonumber\]
Therefore Equation \ref{eq:11.1.12} holds if \(f\equiv1\) and \(g\) is any other function in Equation \ref{eq:11.1.11}.
If \(f(x)=\cos m\pi x/L\) and \(g(x)=\cos n\pi x/L\) where \(m\) and \(n\) are distinct positive integers, then
\[\label{eq:11.1.14} \int_{-L}^L f(x)g(x)\,dx=\int_{-L}^L\cos{m\pi x\over L} \cos{n\pi x\over L}\,dx.\]
To evaluate this integral, we use the identity
\[\cos A\cos B={1\over2}[\cos(A-B)+\cos(A+B)]\nonumber\]
with \(A=m\pi x/L\) and \(B=n\pi x/L\). Then Equation \ref{eq:11.1.14} becomes
\[\int_{-L}^L f(x)g(x)\,dx={1\over2}\left[\int_{-L}^L\cos{(m-n)\pi x\over L}\,dx +\int_{-L}^L\cos{(m+n)\pi x\over L}\,dx\right].\nonumber\]
Since \(m-n\) and \(m+n\) are both nonzero integers, Equation \ref{eq:11.1.13} implies that the integrals on the right are both zero. Therefore Equation \ref{eq:11.1.12} is true in this case.
If \(f(x)=\sin m\pi x/L\) and \(g(x)=\sin n\pi x/L\) where \(m\) and \(n\) are distinct positive integers, then
\[\label{eq:11.1.15} \int_{-L}^L f(x)g(x)\,dx=\int_{-L}^L\sin{m\pi x\over L} \sin{n\pi x\over L}\,dx.\]
To evaluate this integral, we use the identity
\[\sin A\sin B={1\over2}[\cos(A-B)-\cos(A+B)]\nonumber\]
with \(A=m\pi x/L\) and \(B=n\pi x/L\). Then Equation \ref{eq:11.1.15} becomes
\[\int_{-L}^L f(x)g(x)\,dx={1\over2}\left[\int_{-L}^L\cos{(m-n)\pi x\over L}\,dx -\int_{-L}^L\cos{(m+n)\pi x\over L}\,dx\right]=0.\nonumber\]
If \(f(x)=\sin m\pi x/L\) and \(g(x)=\cos n\pi x/L\) where \(m\) and \(n\) are positive integers (not necessarily distinct), then
\[\int_{-L}^L f(x)g(x)\,dx=\int_{-L}^L\sin{m\pi x\over L} \cos{n\pi x\over L}\,dx=0\nonumber\]
because the integrand is an odd function and the limits are symmetric about \(x=0\).
Exercises 11.1.19-11.1.22 ask you to verify that the eigenfunctions of Problems 1-4 are orthogonal on \([0,L]\). However, this also follows from a general theorem that we'll prove in Chapter 13.
