4038T2

# 4038T2 - m =0 a m x m r to 4 xy 2 y y = 0 a Find all values...

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Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books or notes are allowed, but a scientifc calculator is ok - though unnecessary . All work must be shown to receive credit. Please do not give decimal approximations For square roots, For trigonometric or exponential Functions, or For π . Maximum score = 100 points. A list oF useFul Formulas appears below. Legendre’s Equation: (1 - x 2 ) y 0 - 2 xy 0 + n ( n +1) y =0 Bessel’s Equation: x 2 y 0 + xy 0 +( x 2 - ν 2 ) y =0 1. (20) ±ind a non-trivial polynomial solution to (9 - x 2 ) y 0 - 2 xy 0 +20 y =0 . (H in t : substitute x

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2. (30) There are solutions of the form y =

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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 undi±erentiated 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 4-1 4 ) y = 0 (Hint: Substitute z = x 2 .) 4...
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## This note was uploaded on 01/06/2012 for the course MATH 4038 taught by Professor Staff during the Summer '08 term at LSU.

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4038T2 - m =0 a m x m r to 4 xy 2 y y = 0 a Find all values...

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