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Finally, it is necessary to consider the possibility that X may be complex. It is possible to show by direct calculation that the problem (18), (19) has no complex eigenvalues. However, in Section 11.2 we consider in more detail a large class of problems that includes this example. One of the things we show there is that every problem in this class has only real eigenvalues. Therefore we omit the discussion of the nonexistence of complex eigenvalues here. Thus we conclude that all the eigenvalues and eigenfunctions of the problem (18), (19) are given by Eqs. (25) and (26).
PROBLEMS In each of Problems 1 through 6 state whether the given boundary value problem is homogeneous
³ or nonhomogeneous.
1. y" + 4y = 0, y(—1) = 0, y(1) = 0
2. [(1 + x2)/]' + 4 y = 0, y(0) = 0, y(1) = 1
3. Ó + 4y = sinx, y(0) = 0, y(1) = 0
4. — Ó + x2 y = Xy, y (0) — y(0) = 0, /(1) + y(1) = 0
5. —[(1 + x2 )/]' = Xy + 1, y(—1) = 0, y(1) = 0
6. — ó = X(1 + x2 )y, y (0) = 0, /(1) + 3 y(1) = 0
In each of Problems 7 through 10 determine the form of the eigenfunctions and the determinantal equation satisfied by the nonzero eigenvalues. Determine whether X = 0 is an eigenvalue, and
find approximate values for Xj and X2, the nonzero eigenvalues of smallest absolute value.
Estimate Xn for large values of n. Assume that all eigenvalues are real.
7. Ó + Xy = 0, 8. Ó + Xy = 0,
y(0) = 0, y(n) + Ó(ï) = 0 y (0) = 0, y(1) + Ó(1) = 0
9. Ó + Xy = 0, 10. / — Xy = 0,
y(0) — y (0) = 0, y(1) + Ó(\) = 0 y(0) + y (0) = 0, y(1) = 0
11. Consider the general linear homogeneous second order equation
P(x)y+ Q (x) y + R(x) y = 0. (i)
We seek an integrating factor p(x) such that, upon multiplying Eq. (i) by p(x), the resulting equation can be written in the form
[p(x) P(x)/]' + p(x) R(x)y = 0. (ii)
11.1 The Occurrence of Two-Point Boundary Value Problems
(a) By equating coefficients of y, show that ä must be a solution of
PÄ = (Q — Ð)ä.
(b) Solve Eq. (iii) and thereby show that
Compare this result with that of Problem 27 in Section 3.2.
In each of Problems 12 through 15 use the method of Problem 11 to transform the given equation into the form [p(x)y] + q(x)y = 0.
where a2 > 0, c > 0, and k > 0 are constants, is known as the telegraph equation. It arises in the study of an elastic string under tension (see Appendix B of Chapter 10). Equation (i) also occurs in other applications. Assuming that F(x, t) = 0, let u(x, t) = X(x) T(t), separate the variables in Eq. (i), and derive ordinary differential equations for X and T.
17. Consider the boundary value problem
(a) Introduce a new dependent variable u by the relation y = s(x)u. Determine s(x) so that the differential equation for u has no U term.
(b) Solve the boundary value problem for u and thereby determine the eigenvalues and eigenfunctions of the original problem. Assume that all eigenvalues are real.
(c) Also solve the given problem directly (without introducing u).
18. Consider the boundary value problem
(a) Determine, at least approximately, the real eigenvalues and the corresponding eigenfunctions by proceeding as in Problem 17(a, b).
(b) Also solve the given problem directly (without introducing a new variable).
Hint: In part (a) be sure to pay attention to the boundary conditions as well as the differential equation.
The differential equations in Problems 19 and 20 differ from those in previous problems in that the parameter k multiplies the y term as well as the y term. In each of these problems determine the real eigenvalues and the corresponding eigenfunctions.
12. Ó — 2xy + ky = 0,
13. x2/ + x/ + (x2 — v2)y = 0,
14. x/ + (1 — x)y + ky = 0,
15. (1 — x2 )Ó' — xy + a2 y = 0,
Hermite equation Bessel equation Laguerre equation Chebyshev equation
16. The equation
utt + cut + ku = a2 uxx + F(x, t),
y — 2 Ó+ (1 + k) y = 0, y(0) = 0, yd) = 0.