homework 3 solutions addon

Homework 3 - Homework 3 part 2 Solutions and remarks on selected problems Greenberg 4.3 6(a The equation is 2 x 2 y 00 xy x 4 y = 0(We have

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: Homework 3, part 2: Solutions and remarks on selected problems Greenberg, 4.3, 6(a). The equation is 2 x 2 y 00 + xy + x 4 y = 0. (We have multiplied through by an additional factor of x to simplify bookkeeping.) Substitution of ∑ ∞ a n x r + n in the equation leads to ∞ X n =0 [ F ( r + n ) a n + a n- 4 ] x r + n = 0 , (1) where as usual, a- k = 0 for integers k > 0, and where F ( r ) = 2 r ( r- 1) + r = r (2 r- 1). The coefficient corresponding to n = 0 in (1) is set to 0 by choosing r to satisfy the indicial equation F ( r ) = r (2 r- 1) = 0. The two roots are r 1 = 1 / 2 and r 2 = 0. To find a solution corresponding to r 1 set r = 1 / 2 in (1), and set the coefficients for n ≥ 1 equal to zero, to get the recursion equation a n =- 1 F ( n +(1 / 2)) a n- 4 =- 1 2 n ( n + 1 / 2) a n- 1 = 0 , n ≥ 1 . (2) Since a n- 4 = 0 if n < 4, this recursion relation implies a 1 = a 2 = a 3 = 0. Next, taking n = 4 in (2), a 4 =- a / 36. Then a 5 = a 6 = a 7 = 0 because by the recursion formula they are multiples of a 1 , a 2 , and a 3 respectively. Next, for n = 8, a 8 =- a 4 / (136) = a / (36 · 136) = a / 4896. It is clear that only every fourth coefficient will be non-zero. Setting a = 1, y 1 ( x ) = x 1 / 2 1- x 4 36 + x 8 4896 + a 12 x 12 + ··· ! . The recursion relations can be solved explicitly using the Gamma function to get y 1 ( x ) = x 1 / 2 ∞ X k =0 (- 1) k Γ(9 / 8) (32) k k !Γ( k +(9 / 8)) x 4 k . To find a solution corresponding to r 2 set r = 0 in (1), and set the coefficients for n ≥ 1 equal to zero, to get the recursion equation a n =- 1 F ( n ) a n- 4 =- 1 2 n ( n- 1 / 2) a n- 1 = 0 , n ≥ 1 . (3) As before, since a n- 4 = 0 if n < 4, this recursion relation implies a 1 = a 2 = a 3 = 0....
View Full Document

This note was uploaded on 09/29/2009 for the course 650 527 taught by Professor Danocone during the Fall '06 term at Rutgers.

Page1 / 4

Homework 3 - Homework 3 part 2 Solutions and remarks on selected problems Greenberg 4.3 6(a The equation is 2 x 2 y 00 xy x 4 y = 0(We have

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