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

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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....
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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.

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

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