# hw7 - 5.2.1 Consider y 00 y = 0 x = 0 We seek a series...

This preview shows pages 1–8. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5.2.1. Consider y 00- y = 0, x = 0. We seek a series solution about x = 0. i.e. We are looking for a solution of the form y ( x ) = ∞ X n =0 a n x n . (1) It follows then that y 00 ( x ) = ∞ X n =0 n ( n- 1) a n x n- 2 = ∞ X n =2 n ( n- 1) a n x n- 2 . (The last equality follows from the fact that the first two terms of the series were 0.) Shifting indices to start from n = 0, we have y 00 ( x ) = ∞ X n =0 ( n + 2)( n + 1) a n +2 x n . (2) Plugging (1) and (2) into our differential equation, we see ∞ X n =0 (( n + 2)( n + 1) a n +2- a n ) x n = 0 . Thus as the equality above must hold for all x , ( n + 2)( n + 1) a n +2- a n = 0 for all n . That is, we have the recurrence relation a n +2 = a n ( n + 1)( n + 2) . Note for n even, a n = a n- 2 ( n- 1) n = a n- 4 ( n- 3)( n- 2)( n- 1) n = ··· = a 1 · 2 ··· n = a n ! . Similarly for n odd, a n = a n- 2 ( n- 1) n = a n- 4 ( n- 3)( n- 2)( n- 1) n = ··· = a 1 2 · 3 ··· n = a 1 n ! . We use theorem 5.3.1 to find a lower bound for the radius of convergence. In the language of the theorem, p = 0 = ∑ x n and q =- 1 =- 1 + ∑ x n . These trivial series clearly have an infinite radius of convergence, and so our series has a radius of convergence of at least the minimum of ∞ and ∞ . i.e. We have an infinite radius of convergence. We are free to vary a and a 1 as we please. A reasonable way to get two linearly independent solutions is to first take a = 1 and a 1 = 0 and then to take a = 0 and a 1 = 1....
View Full Document

{[ snackBarMessage ]}

### Page1 / 11

hw7 - 5.2.1 Consider y 00 y = 0 x = 0 We seek a series...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online