Tutorial 3 - MATH3705 Tutorial 3 1. Consider the DE: y +...

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1. Consider the DE: y ′′ + 2 xy + y = 0. (a) Find the general solution of the equation about x 0 = 0. (b) Solve the initial-value problem if y (0) = 2 and y (0) = 3. Solution: (a) 1) Consider singularities: none. So we shall attempt to find a solution in the form of an infinite series. 2) Let y = n =0 a n x n . Then y = ± n =1 na n x n 1 , xy = ± n =1 na n x n , and y ′′ = ± n =2 n ( n 1) a n x n 2 = ± n =0 ( n + 2)( n + 1) a n x n . 3) We imply that a n +2 = 2 n + 1 ( n + 1)( n + 2) a n , which gives, a 2 n = ( 1) n 1 ··· 5 · 9 ··· (4 n 4) (2 n )! a 0 , a 2 n +1 = ( 1) n 3 ··· 7 · 11 ··· (4 n 1) (2 n + 1)! a 1 . 4) We have y = a 0 ( 1 1 2 x 2 + ± n =2 ( 1) n 1 ··· 5 · 9 ··· (4 n 4) (2 n )! x 2 n ) + a 1 ( x + ± n =1 ( 1) n 3 ··· 7 · 11 ··· (4 n 1) (2 n + 1)! x
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This note was uploaded on 07/11/2011 for the course ECE 45 taught by Professor Lee during the Spring '08 term at Alfred University.

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Tutorial 3 - MATH3705 Tutorial 3 1. Consider the DE: y +...

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