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Unformatted text preview: GE 207K Series solutions near Singular points – PS October 30, 2010 Problem 1 For the following differential equation 2 xy 00 + y + xy = 0 , (1) show that x = 0 is a regular singular point, and find the series solution corresponding to the largest root at the singularity. Solution : In this problem, P ( x ) = 2 x , and P ( x = 0) = 0 and so x = 0 is a singular point. Next we verify that x is a regular singular point: lim x → x 1 2 x = 1 2 < ∞ , X lim x → x 2 x 2 x = 0 < ∞ . X Since x = 0 is a regular singular point, we have at least one solution in the form of y = x r ∑ ∞ n =0 x n = ∑ ∞ n =0 x n + r . Differentiating the solution form (with respect to x ) we have: ! Remember, n = 0 after you differ- entiate! y = ∞ X n = ( n + r ) a n x n + r- 1 , (2) y 00 = ∞ X n = ( n + r )( n + r- 1) a n x n + r- 2 . (3) Next, plug these back into the differential equation: 2 x ∞ X n = ( n + r )( n + r- 1) a n x n + r- 2 + ∞ X n = ( n + r ) a n x n + r- 1 + x ∞ X n =0 a n x n + r = 0 . (4) Following the steps we talked about, we shouldn’t have any terms multiplied by the sum- mations. Therefore, distribute 2...
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This note was uploaded on 05/02/2011 for the course GE 207K taught by Professor None during the Spring '10 term at University of Texas.
- Spring '10