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

Info iconThis preview shows pages 1–2. 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 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 shouldnt have any terms multiplied by the sum- mations. Therefore, distribute 2...
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

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