3 - kbsing - March 26, 2010 The differential equation of...

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: March 26, 2010 The differential equation of interest is: ln x y 00 + 1 2 y + y = 0 . Determine the first three non-zero terms in the series ∑ ∞ n =0 a n ( x- 1) n + r First verify that x = 1 is indeed a singular point (show this) : lim x → 1 1 2 ( x- 1) ln x = 1 2 < ∞ lim x → 1 " ( x- 1) 2 ln x # = 0 < ∞ ∴ ( x = 1) is a singular point. Also, there exists at least one solution of the form y = ( x- 1) r ∞ X n =0 a n ( x- 1) n = ∞ X n =0 a n ( x- 1) n + r . (1) Aside: You can use a change of variables t = x- 1 to re-write the differential equation in terms of t and move the singular point to t = 0 : let t = x- 1 ⇒ dy dx = dy dt dt dx = dy dt · 1 = y ( t ) ⇒ ln xy 00 ( x ) + 1 2 y ( x ) + y ( x ) = ln ( t + 1) y 00 ( t ) + 1 2 y ( t ) + y ( t ) = 0 Then the solution appears in form of y ( t ) = t r ∞ X n =0 ˆ a n t n = ∞ X n =0 ˆ a n t n + r . This will make the expressions appear more compact and is definitely a good idea to do. In fact, do this. But I’ll keepfact, do this....
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.

Page1 / 3

3 - kbsing - March 26, 2010 The differential equation of...

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