2 - singular_ps - GE 207K Series solutions near Singular...

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GE 207K Series solutions near Singular points – PS October 30, 2010 Problem 1 For the following differential equation 2 xy 00 + y 0 + xy = 0 , (1) show that x 0 = 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) = 0 and so x 0 = 0 is a singular point. Next we verify that x 0 is a regular singular point: lim x 0 x 1 2 x = 1 2 < , X lim x 0 x 2 x 2 x = 0 < . X Since x 0 = 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 0 = X n = 0 ( n + r ) a n x n + r - 1 , (2) y 00 = X n = 0 ( n + r )( n + r - 1) a n x n + r - 2 . (3) Next, plug these back into the differential equation: 2 x X n = 0 ( n + r )( n + r - 1) a n x n + r - 2 + X n = 0 ( 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 x into the first summation, and distribute x into the third
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