pset_7 - leading coefficient of y 00 is just a...

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ACM 95/100b Problem Set 7 March 4, 2009 Due by 5:00PM on 3/11/2009 All problems are worth 10 points on this set. Please deposit your problem set in the slot in 303 Firestone or upload it via Moodle. In either case please keep a copy of your problem set. Please remember to include your section number and section instructor. Collaboration is allowed on all problems but please write up the solutions yourself. Problem 1 Consider the ODE d dx ± (1 - x 2 ) dy dx ² + λy = x 2 over the interval - 1 x 1 with boundary conditions y ( - 1) finite y (1) finite Solve this problem using a series of Legendre polynomials. That is represent y ( x ) = X n =0 A n P n ( x ) and solve for the A n . Does the solution exist for all values of λ ? Problem 2 Consider the Chebyshev ODE given by d dx ± 1 - x 2 dy dx ² + λy ( x ) 1 - x 2 = 0 - 1 x 1 (a) Classify the singular points at x = ± 1 1
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(b) Using a Taylor series about x = 0 generate Taylor series for two linearly inde- pendent solutions. Hint: it may be advantageous to rewrite the ODE so that the
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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.

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pset_7 - leading coefficient of y 00 is just a...

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