# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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1. Find a formal solution of the nonhomogeneous boundary value problem

- (xy) = V-xy + f (x), y, y bounded as x ^ 0, y(1) = 0,

where f is a given continuous function on 0 < x < 1, and p, is not an eigenvalue of the corresponding homogeneous problem.

Hint: Use a series expansion similar to those in Section 11.3.

2. Consider the boundary value problem

- (xy)' = ^xy,

y, y bounded as x ^ 0, y (1) = 0.

(a) Show that k0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction 00(x) = 1. If k > 0, show formally that the eigenfunctions are given by (x) =

662

Chapter 11. Boundary Value Problems and Sturm Liouville Theory

I

J0(^/k~x), where ..Jk~n is the nth positive root (in increasing order) of the equation J (\fk) = 0. It is possible to show that there is an infinite sequence of such roots.

(b) Show that if m, n = 0, 1, 2,..., then 1

x<pm(x)0n(x) dx = 0, m = n.

0

(c) Find a formal solution to the nonhomogeneous problem

- (xf)' = pxy + f (x),

y, y bounded as x ^ 0, y (1) = 0,

where f is a given continuous function on 0 < x < 1, and p is not an eigenvalue of the

corresponding homogeneous problem.

3. Consider the problem

- (x/)' + (k2/x)y = kxy, y, y bounded as x ^ 0, y(1) = 0,

where k is a positive integer.

(a) Using the substitution t = -Jk x, show that the given differential equation reduces to Bessel’s equation of order k (see Problem 9 of Section 5.8). One solution is Jk(t); a second linearly independent solution, denoted by Yk(t), is unbounded as t ^ 0.

(b) Show formally that the eigenvalues k1,k2,... of the given problem are the squares of the positive zeros of Jk j/k), and that the corresponding eigenfunctions are 0n (x) = Jk(yfk~x). It is possible to show that there is an infinite sequence of such zeros.

(c) Show that the eigenfunctions 0n (x) satisfy the orthogonality relation

-1

(x)$„(x) dx = 0, m = n.

/0

(d) Determine the coefficients in the formal series expansion

J0

n= 1

(e) Find a formal solution of the nonhomogeneous problem

- (xy) + (k2/x)y = pxy + f(x),

y, y bounded as x ^ 0, y(1) = 0,

where f is a given continuous function on 0 < x < 1, and p is not an eigenvalue of the corresponding homogeneous problem.

4. Consider Legendre’s equation (see Problems 22 through 24 of Section 5.3)

-[(1 - x2)/]' = ky

subject to the boundary conditions

y(0) = 0, y, y bounded as x ^ 1.

The eigenfunctions of this problem are the odd Legendre polynomials 0 (x) = P1 (x) = x, 02(x) = P3(x) = (5x3 - 3x)/2,..., 0(x) = P, j(x),... corresponding to the eigenvalues Y = 2, k2 = 4 • 3,...,kn = 2n(2n - 1),....

(a) Show that

1

1

(x)$„(x) dx = 0, m = n.

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 663

(b) Find a formal solution of the nonhomogeneous problem

— [(1 — x2)/]' = py + f (x),

y(0) = 0, y, y bounded as x ^ 1,

where f is a given continuous function on 0 < x < 1, and p is not an eigenvalue of the corresponding homogeneous problem.

5. The equation

(1 — x2)y" — x/ + ky = 0 (i)

is Chebyshev’s equation; see Problem 10 of Section 5.3.

(a) Show that Eq. (i) can be written in the form

— [(1 — x2)1/2/]' = k(1 — x2)—1/2y, —1 < x < 1. (ii)

(b) Consider the boundary conditions

y, / bounded as x ^ —1, y, / bounded as x ^ 1. (iii)

Show that the boundary value problem (ii), (iii) is self-adjoint.

(c) It can be shown that the boundary value problem (ii), (iii) has the eigenvalues k0 = 0, k1 = 1, k2 = 4,...,kn = n2, .... The corresponding eigenfunctions are the Chebyshev polynomials Tn(x): T0(x) = 1, T1 (x) = x, T2(x) = 1 — 2x2, .... Show that

Tm(x)Tn(x) . 0 , )

m n dx = 0, m = n. (iv)

/1

-1 (1 — x2)1/2 Note that this is a convergent improper integral.

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion

In this chapter we are interested in extending the method of separation of variables developed in Chapter 10 to a larger class of problems—to problems involving more general differential equations, more general boundary conditions, or different geometrical regions. We indicated in Section 11.3 how to deal with a class of more general differential equations or boundary conditions. Here we concentrate on problems posed in various geometrical regions, with emphasis on those leading to singular Sturm-Liouville problems when the variables are separated.

Because of its relative simplicity, as well as the considerable physical significance of many problems to which it is applicable, the method of separation of variables merits its important place in the theory and application of partial differential equations. However, this method does have certain limitations that should not be forgotten. In the first place, the problem must be linear so that the principle of superposition can be invoked to construct additional solutions by forming linear combinations of the fundamental solutions of an appropriate homogeneous problem.

As a practical matter, we must also be able to solve the ordinary differential equations, obtained after separating the variables, in a reasonably convenient manner. In some problems to which the method of separation of variables can be applied in principle, it

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