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Unformatted text preview: m =0 a m x m + r to 4 xy + 2 y + y = 0. a. Find all values of r which yield a solution. b. Find a recursion formula for a m +1 in terms of a m , m , and r . c. Now set r equal to its smallest possible value from part (a), and nd a formula for a m in terms of m . 2 3. (25) (1 + x ) x 2 y(1 + 2 x ) xy + (1 + 2 x ) y = 0 has a solution y 1 = x + C . a. Find the value of C which makes y 1 a solution. b. Find a solution y 2 = y 1 u which is independent of y 1 . (Hint: What must the sum of all the terms which have an undierentiated u as a factor?) 3 4. (25) Find the general solution for y in terms of x using Bessel functions: x 2 y + xy + (4 x 41 4 ) y = 0 (Hint: Substitute z = x 2 .) 4...
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 Summer '08
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