1 - singular_steps - GE 207K Series Solutions Near a...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: GE 207K Series Solutions Near a Singular Point March 10, 2011 This page aims to recap our mini-lecture 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 ....
View Full Document

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.

Page1 / 6

1 - singular_steps - GE 207K Series Solutions Near a...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online