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Unformatted text preview: Here is the idea: substitute y = ∑ n = ∞ c n x n into the differential equation and then equate all coefficients to the righthand side of the equation to determine the coefficients c n . Example 1. Solve Slution. Let y = ∑ n = ∞ c n x n . Then y' = ∑ n = 1 ∞ nc n x n − 1 and y' ' = ∑ n = 2 ∞ n ( n − 1 ) c n x n − 2 . Substituting y ∧ y ' ' into differential equation gives ¿ 2 c 2 + ∑ n = 3 ∞ n ( n − 1 ) c n x n − 2 + ∑ n = ∞ c n x n + 1 Let k = n − 2 ∧ k = n + 1 . Then c k − 1 x k = ¿ 2 c 2 + ∑ k = 1 ∞ ( k + 2 )( k + 1 ) c k + 2 x k + ∑ k = 1 ∞ ¿ This relation generates consecutive coefficients of the assumed solution one at the time as we let k take on the successive integers Example 2. Use power series method to solve given differential equation about ordinary point x = 0, y '' +( cosx ) y = . Solution....
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 Fall '15
 Linear Algebra, Algebra, Power Series, 0 k, 1 k

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