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Unformatted text preview: GE 207K Series Solutions Near a Singular Point March 10, 2011 This page aims to recap our minilecture on series solution near a Singular point. We wish expand the solutions of a differential equation that appears as P ( x ) y 00 + Q ( x ) y 00 + R ( x ) y = 0 , (1) about a point of interest, labeled hereon as x . We call this point an singular point if P ( x ) = 0 . For a more rigorous definition of an singular point, refer to Section 5.3 of your book. Problem statement: Find (at least one) series solution of the differential equation P ( x ) y 00 + Q ( x ) y + R ( x ) y = 0 , (2) about x . Steps: 1. Make sure that x is a regular singular point. If both of the following inequalities hold, then x is a regular singular point. p = lim x x ( x x ) Q ( x ) P ( x ) < , (3) q = lim x x ( x x ) 2 R ( x ) P ( x ) < . (4) 2. If x is a regular singular point, we have at least one solution in the form of y = x r X n =0 a n x n = X n =0 a n x n + r = a x r + a 1 x 1+ r + a 2 x 2+ r + . (5) Take the derivative of the solution form: ! We cannot change the index n here. y = X n = ( n + r ) a n x n + r 1 , y 00 = X n = ( n + r )( n + r 1) a n x n + r 2 . (6) 3. Plug y , y , y 00 back into the DE P ( x ) X n = ( n + r ) ( n + r 1) a n x n + r 2 + Q ( x ) X n = ( n + r ) a n x n + r 1 + R ( x ) X n =0 a n x n + r = 0 1 GE 207K Series Solutions Near a Singular Point March 10, 2011 4. Move all the coefficients P ( x ) ,Q ( x ) , and R ( x ) into the summations. 5. Check the lengths of all series. That is, make the first term of all summations start from the same power of x ....
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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 at Austin.
 Spring '10
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