s09_mthsc208_hw16

s09_mthsc208_hw16 - MTHSC 208 (Differential Equations) Dr....

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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 16 Due Monday March 23rd, 2009 (1) Use the ratio test to find the radius of convergence of the following power series: ∞ 1 (x − π )n , n+1 n=0 b. n=0 ∞ d. ∞ ∞ (−1)n xn , a. 1 (x − π )n , 2n n=0 c. ∞ ∞ (5x − 10)n , e. 3 (x − 2)n , n+1 n=0 f. n=0 1 (3x − 6)n . n! n=0 (2) Use the comparison test to find an estimate for the radius of convergence of each of the following power series: ∞ a. ∞ 1 x2n , (2n)! n=0 (−1)n x2n , b. n=0 ∞ ∞ c. 1 (x − 4)2n , 2n n=1 1 (x − π )2n . 22 n n=0 d. (3) Find the radius of convergence of the power series ∞ (−1)n n=0 1 (n!)2 x 2 2n . (4) The differential equation dy d2 y −x + p2 y = 0, 2 dx dx where p is a constant, is known as Chebyshev’s equation. It can be rewritten in the form (1 − x2 ) d2 y dy x p2 − P (x) + Q(x)y = 0, where P (X ) = − , Q(x) = . 2 2 dx dx 1−x 1 − x2 (a) If P (x) and Q(x) are represented as a power series about x0 = 0, what is the radius of convergence of these power series? (b) Assuming a power series centered at 0, find the recursion formula for an+2 in terms of an . (c) Use the recursion formula to determine an in terms of a0 and a1 , for 2 ≤ n ≤ 9. (d) In the special case where p = 3, find a nonzero polynomial solution to this differential equation. ...
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