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Unformatted text preview: leading coefﬁcient of y 00 is just a polynomial (c) Show that it is only possible to get bounded solutions at x = ± 1 if λ = n 2 with n = 0 , 1 , 2 ,.... and that these solutions are polynomials (known as the Chebyshev polynomials T n ( x ) ). Determine the ﬁrst three polynomials. (d) Using the substitution x = cos ξ show that a closed form expression for the Chebyshev polynomials is T n ( x ) = cos( n cos-1 x ) Problem 3 Using Fourier transforms solve the following boundary value problem y 00-βy = cos(2 x )- ∞ < x < ∞ with boundary conditions y ( x ) ﬁnite as | x | → ∞ . Here β is real and positive. Problem 4 Using an appropriate transform solve the boundary value problem y 00-y = sin( x ) < x < ∞ with boundary conditions y (0) = 1 y ( x ) bounded as x → + ∞ 2...
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This note was uploaded on 12/12/2009 for the course ACM 95b taught by Professor Nilesa.pierce during the Winter '09 term at Caltech.
- Winter '09