6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan z
6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan z
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:6. Consider the eigenvalue problem
y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0.
All the eigenvalues are nonnegative, so write à = a²
where a 2 0. (a) Show that A = 0 is not an eigen-
value. (b) Show that y = Acos ax + B sin ax satis-
fies the endpoint conditions if and only if B = 0 and a
is a positive root of the equation tan z = 1/z. These roots
{an}° are the abscissas of the points of intersection of the
curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13.
Thus the eigenvalues and eigenfunctions of this problem
are the numbers {a;}° and the functions {cos an x}9°, re-
spectively.
y =
2n
Зл
I ly = tan z
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