INTRODUCTION

For convenience we present here a brief review of some of the ordinary differential equations that will be of importance in the sections and chapters that follow.

Linear equations	General solutions
Cauchy–Euler equation	General solutions, x > 0
Parametric Bessel equation (ν = 0)	General solution, x > 0
Legendre’s equation (n = 0, 1, 2, …)	Particular solutions are polynomials

Regarding the two forms of the general solution of y″ – α²y = 0, we will, in the future, employ the following informal rule:

Eigenvalues and Eigenfunctions

Orthogonal functions arise in the solution of differential equations. More to the point, an orthogonal set of functions can be generated by solving a two-point boundary-value problem involving a linear second-order differential equation containing a parameter λ. In Example 2 of Section 3.9 we saw that the boundary-value problem

y″ + λy = 0, y(0) = 0, y(L) = 0, (1)

possessed nontrivial solutions only when the parameter λ took on the values λ_n = n²π²/L², n = 1, 2, 3, … called eigenvalues. The corresponding nontrivial solutions y = c₂ sin(nπx/L) or simply y = sin(nπx/L) are called the eigenfunctions of the problem. For example, for (1) we have

For our purposes in this chapter it is important to recognize the set of functions generated by this BVP; that is, {sin(nπx/L)}, n = 1, 2, 3, …, is the orthogonal set of functions on the interval [0, L] used as the basis for the Fourier sine series.

EXAMPLE 1 Eigenvalues and Eigenfunctions

Regular Sturm–Liouville Problem

The problems in (1) and (2) are special cases of an important general two-point boundary-value problem. Let p, q, r, and r′ be real-valued functions continuous on an interval [a, b], and let r(x) > 0 and p(x) > 0 for every x in the interval. Then

Solve: [r(x)y′] + (q(x) + λp(x))y = 0(3)

Subject to: A₁y(a) + B₁y′(a) = 0(4)

A₂ y(b) + B₂ y′(b) = 0(5)

is said to be a regular Sturm–Liouville problem. Jacques Charles François Sturm (1803–1855) was born in Switzerland but spent his adult life as a professor of mathematics and mechanics in Paris, France. Sturm’s name is one of 72 names of scientists, engineers, and mathematicians engraved on the Eiffel Tower. The French mathematician Joseph Liouville (1809–1882) was also a professor in Paris. In addition to his work with Sturm on boundary-value problems, an important theorem in complex analysis bears his name (Liouville’s theorem). The small impact crater Liouville on the moon is named after him.

The coefficients in the boundary-conditions (4) and (5) are assumed to be real and independent of λ. In addition, A₁ and B₁ are not both zero, and A₂ and B₂ are not both zero. The boundary-value problems in (1) and (2) are regular Sturm–Liouville problems. From (1) we can identify r(x) = 1, q(x) = 0, and p(x) = 1 in the differential equation (3); in boundary condition (4) we identify a = 0, A₁ = 1, B₁ = 0, and in (5), b = L, A₂ = 1, B₂ = 0. From (2) the identifications would be a = 0, A₁ = 0, B₁ = 1 in (4), and b = L, A₂ = 0, B₂ = 1 in (5).

The differential equation (3) is linear and homogeneous. The boundary conditions in (4) and (5), both a linear combination of y and y′ equal to zero at a point, are also called homogeneous. A boundary condition such as A₂ y(b) + B₂ y′(b) = C₂, where C₂ is a nonzero constant, is nonhomogeneous. Naturally, a boundary-value problem that consists of a homogeneous linear differential equation and homogeneous boundary conditions is said to be homogeneous; otherwise it is nonhomogeneous. The boundary conditions (4) and (5) are said to be separated because each condition involves only a single boundary point. Boundary conditions are referred to as mixed if each condition involves both boundary points x = a and x = b. For example, the periodic boundary conditions y(a) = y(b), y′(a) = y′(b) are mixed boundary conditions.

Because a regular Sturm–Liouville problem is a homogeneous BVP, it always possesses the trivial solution y = 0. However, this solution is of no interest to us. As in Example 1, in solving such a problem we seek numbers λ (eigenvalues) and nontrivial solutions y that depend on λ (eigenfunctions).

Properties

Theorem 12.5.1 is a list of some of the more important of the many properties of the regular Sturm–Liouville problem. We shall prove only the last property.

PROOF OF (d):

Let y_m and y_n be eigenfunctions corresponding to eigenvalues λ_m and λ_n, respectively. Then

(6)

(7)

Multiplying (6) by y_n and (7) by y_m and subtracting the two equations gives

Integrating this last result by parts from x = a to x = b then yields

(8)

Now the eigenfunctions y_m and y_n must both satisfy the boundary conditions (4) and (5). In particular, from (4) we have

A₁y_m(a) + B₁(a) = 0

A₁y_n(a) + B₁(a) = 0.

For this system to be satisfied by A₁ and B₁, not both zero, the determinant of the coefficients must be zero:

y_m(a)(a) – y_n(a)(a) = 0.

A similar argument applied to (5) also gives

y_m(b)(b) – y_n(b)(b) = 0.

Using these last two results in (8) shows that both members of the right-hand side are zero. Hence we have established the orthogonality relation

p(x)y_m(x)y_n(x) dx = 0, λ_m ≠ λ_n. (9)≡

It can also be proved that the orthogonal set of eigenfunctions {y₁(x), y₂(x), y₃(x), …} of a regular Sturm–Liouville problem is complete on [a, b]. See page 684.

EXAMPLE 2 A Regular Sturm–Liouville Problem

Solve the boundary-value problem

y″ + λy = 0, y(0) = 0, y(1) + y′(1) = 0.(10)

SOLUTION

You should verify that for λ = 0 and for λ = –α² < 0, where α > 0, the BVP in (10) possesses only the trivial solution y = 0. For λ = α² > 0, α > 0, the general solution of the differential equation y″ + α²y = 0 is y = c₁cos αx + c₂sin αx. Now the condition y(0) = 0 implies c₁ = 0 in this solution and so we are left with y = c₂ sin αx. The second boundary condition y(1) + y′(1) = 0 is satisfied if

Choosing c₂ ≠ 0, we see that the last equation is equivalent to

(11)

If we let x = α in (11), then FIGURE 12.5.1 shows the plausibility that there exists an infinite number of roots of the equation tan x = –x, namely, the x-coordinates of the points where the graph of y = –x intersects the branches of the graph of y = tan x. The eigenvalues of problem (10) are then , where α_n, n = 1, 2, 3, …, are the consecutive positive roots α₁, α₂, α₃, … of (11). With the aid of a CAS it is easily shown that, to four rounded decimal places, α₁ = 2.0288, α₂ = 4.9132, α₃ = 7.9787, and α₄ = 11.0855, and the corresponding solutions are y₁ = sin 2.0288x, y₂ = sin 4.9132x, y₃ = sin 7.9787x, and y₄ = sin 11.0855x. In general, the eigenfunctions of the problem are {sin α_n x}, n = 1, 2, 3, ….

With identifications r(x) = 1, q(x) = 0, p(x) = 1, A₁ = 1, B₁ = 0, A₂ = 1, and B₂ = 1 we see that (10) is a regular Sturm–Liouville problem. Thus {sin α_n x}, n = 1, 2, 3, … is an orthogonal set with respect to the weight function p(x) = 1 on the interval [0, 1].≡

In some circumstances we can prove the orthogonality of the solutions of (3) without the necessity of specifying a boundary condition at x = a and at x = b.

Singular Sturm–Liouville Problem

There are several other important conditions under which we seek nontrivial solutions of the differential equation (3):

• r(a) = 0 and a boundary condition of the type given in (5) is specified at x = b;(12)
• r(b) = 0 and a boundary condition of the type given in (4) is specified at x = a;(13)
• r(a) = r(b) = 0 and no boundary condition is specified at either x = a or at x = b;(14)
• r(a) = r(b) and boundary conditions y(a) = y(b), y′(a) = y′(b).(15)

The differential equation (3) along with one of conditions (12) or (13) is said to be a singular boundary-value problem. Equation (3) with the conditions specified in (15) is said to be a periodic boundary-value problem because the boundary conditions are periodic. Observe that if, say, r(a) = 0, then x = a may be a singular point of the differential equation, and consequently a solution of (3) may become unbounded as x → a. However, we see from (8) that if r(a) = 0, then no boundary condition is required at x = a to prove orthogonality of the eigenfunctions provided these solutions are bounded at that point. This latter requirement guarantees the existence of the integrals involved. By assuming the solutions of (3) are bounded on the closed interval [a, b] we can see from inspection of (8) that

• If r(a) = 0, then the orthogonality relation (9) holds with no boundary condition at x = a;(16)
• If r(b) = 0, then the orthogonality relation (9) holds with no boundary condition at x = b*;(17)
• If r(a) = r(b) = 0, then the orthogonality relation (9) holds with no boundary conditions specified at either x = a or x = b;(18)
• If r(a) = r(b), then the orthogonality relation (9) holds with the periodic boundary conditions y(a) = y(b), y′(a) = y′(b).(19)

Self-Adjoint Form

If we carry out the differentiation , the differential equation in (3) is the same as

(20)

For example, Legendre’s differential equation is exactly of the form given in (20) with and . In other words, another way of writing Legendre’s DE is

(21)

But if you compare other second-order DEs (say, Bessel’s equation, Cauchy–Euler equations, and DEs with constant coefficients) you might believe, given the coefficient of y′ is the derivative of the coefficient of y″, that few other second-order DEs have the form given in (3). On the contrary, if the coefficients are continuous and a(x) ≠ 0 for all x in some interval, then any second-order differential equation

a(x)y″ + b(x)y′ + (c(x) + λd(x))y = 0(22)

can be recast into the so-called self-adjoint form (3). To see this, we proceed as in Section 2.3 where we rewrote a linear first-order equation in the form by dividing the equation by a₁(x) and then multiplying by the integrating factor where, assuming no common factors, P(x) = a₀(x)/a₁(x). So first, we divide (22) by a(x). The first two terms are then where, for emphasis, we have written Y = y′. Second, we multiply this equation by the integrating factor , where a(x) and b(x) are assumed to have no common factors

In summary, by dividing (22) by a(x) and then multiplying by we get

(23)

Equation (23) is the desired form given in (20) and is the same as (3):

For example, to express 3y″ + 6y′ + λy = 0 in self-adjoint form, we write y″ + 2y′ + λ y = 0 and then multiply by . The resulting equation is

It is certainly not necessary to put a second-order differential equation (22) into the self-adjoint form (3) in order to solve the DE. For our purposes we use the form given in (3) to determine the weight function p(x) needed in the orthogonality relation (9). The next two examples illustrate orthogonality relations for Bessel functions and for Legendre polynomials.

EXAMPLE 3 Parametric Bessel Equation

For any choice of A₂ and B₂, not both zero, it is known that (25) has an infinite number of roots x_i = α_i b. The eigenvalues are then λ_i = = (x_i /b)². More will be said about eigenvalues in the next chapter.

EXAMPLE 4 Legendre’s Equation

12.5 Exercises Answers to selected odd-numbered problems begin on page ANS-32.

In Problems 1 and 2, find the eigenfunctions and the equation that defines the eigenvalues for the given boundary-value problem. Use a CAS to approximate the first four eigenvalues λ₁, λ₂, λ₃, and λ₄. Give the eigenfunctions corresponding to these approximations.

1. y″ + λy = 0, y′(0) = 0, y(1) + y′(1) = 0
2. y″ + λy = 0, y(0) + y′(0) = 0, y(1) = 0
3. Consider y″ + λy = 0 subject to y′(0) = 0, y′(L) = 0. Show that the eigenfunctions are

   .

   This set, which is orthogonal on [0, L], is the basis for the Fourier cosine series.

4. Consider y″ + λy = 0 subject to the periodic boundary conditions y(–L) = y(L), y′(–L) = y′(L). Show that the eigenfunctions are

   

   This set, which is orthogonal on [–L, L], is the basis for the Fourier series.

5. Find the square norm of each eigenfunction in Problem 1.
6. Show that for the eigenfunctions in Example 2,

   

7. 1. Find the eigenvalues and eigenfunctions of the boundary-value problem

      x²y″ + xy′ + λy = 0, y(1) = 0, y(5) = 0.

   2. Put the differential equation in self-adjoint form.
   3. Give an orthogonality relation.
8. 1. Find the eigenvalues and eigenfunctions of the boundary-value problem

      y″ + y′ + λy = 0, y(0) = 0, y(2) = 0.

   2. Put the differential equation in self-adjoint form.
   3. Give an orthogonality relation.
9. Laguerre’s differential equation

   

   has polynomial solutions L_n(x). Put the equation in self-adjoint form and give an orthogonality relation on (0, ).

10. Hermite’s differential equation

    

    has polynomial solutions H_n(x). Put the equation in self-adjoint form and give an orthogonality relation on (–, ).

11. Consider the regular Sturm–Liouville problem:

    

    1. Find the eigenvalues and eigenfunctions of the boundary-value problem. [Hint: Let x = tan θ and then use the Chain Rule.]
    2. Give an orthogonality relation.
12. 1. Find the eigenfunctions and the equation that defines the eigenvalues for the boundary-value problem

       

       y is bounded at x = 0, y(3) = 0.

    2. Use Table 5.3.1 of Section 5.3 to find the approximate values of the first four eigenvalues λ₁, λ₂, λ₃, and λ₄.

Discussion Problem

13. Consider the special case of the regular Sturm–Liouville problem on the interval [a, b]:

    

    Is λ = 0 an eigenvalue of the problem? Defend your answer.

Computer Lab Assignments

14. 1. Give an orthogonality relation for the Sturm–Liouville problem in Problem 1.
    2. Use a CAS as an aid in verifying the orthogonality relation for the eigenfunctions y₁ and y₂ that correspond to the first two eigenvalues λ₁ and λ₂, respectively.
15. 1. Give an orthogonality relation for the Sturm–Liouville problem in Problem 2.
    2. Use a CAS as an aid in verifying the orthogonality relation for the eigenfunctions y₁ and y₂ that correspond to the first two eigenvalues λ₁ and λ₂, respectively.

*Conditions (16) and (17) are equivalent to choosing A₁ = 0, B₁ = 0 in (4), and A₂ = 0, B₂ = 0 in (5), respectively.