The equation (1 x2) y__ xy_ + y = 0 (25) is Chebyshevs equation; see of Section 5.3. a
Chapter 11, Problem 5(choose chapter or problem)
The equation (1 x2) y__ xy_ + y = 0 (25) is Chebyshevs equation; see of Section 5.3. a. Show that equation (25) can be written in the form __1 x2_1/2 y__ _ = _1 x2_ 1/2 y, 1 < x < 1. (26) b. Consider the boundary conditions y and y_ remain bounded as x 1+, (27) y and y_ remain bounded as x 1 . Show that the boundary value problem (26), (27) is self-adjoint. c. It can be shown that the singular Sturm-Liouville boundary value problem (26), (27) has eigenvalues 0 = 0, 1 = 1, 2 = 4, . . . , n = n2, . . . . The corresponding eigenfunctions are the Chebyshev polynomials Tn( x): T0( x) = 1, T1( x) = x, T2( x) = 1 2x2, . . . . Show that _ 1 1 Tm( x)Tn( x) _1 x2_1/2 dx = 0, m _= n. (28) Note that this is a convergent improper integral.
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