m238f02t3hs - Mathematics 238 test 3 due Monday note...

Info iconThis preview shows pages 1–3. 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: Mathematics 238 test 3 due Monday, November 25, 2002 note : on-line references to power series may be found on-line at http://www.sosmath.com/diffeq/series/series03/series03.html http://college.hmco.com/mathematics/larson/calculus early/3e/shared/chapter15/clc7eap1505.pdf I found many more references by typing “power series differential equations”into the google search engine (http://www.google.com). (1) Given the differential equation y 00 + ty + 4 y = 0 (a) Find the recursion relation for the power series solution about t = 0. y = ∞ X k =0 a k t k y = ∞ X k =1 ka k t k- 1 and ty = ∞ X k =1 ka k t k y 00 = ∞ X k =2 k ( k- 1) a k t k- 2 = ∞ X k =0 ( k + 2)( k + 1) a k +2 t k therefore y 00 + ty + 4 y = (2 a 2 + 4 a ) + ∞ X k =1 (( k + 2)( k + 1) a k +2 + ka k + 4 a k ) t k = 0 Set the constant coefficient equal to zero: 2 a 2 + 4 a = 0 or a 2 =- 2 a Set the coefficient of t k equal to 0 to get the recursion relation: ( k +2)( k +1) a k +2 +( k +4) a k = 0 or a k +2 =- k + 4 ( k + 2)( k + 1) a k for k ≥ 1 (b) For which values of t will power series solutions of this differential equation converge? According to Fuch’s Theorem, the power series will converge where the coefficient functions have convergent power series. The coefficients t and 4 are themselves simple power series (polynomials) which converge for all values of t. The power series solution hinted at in part (a) converges for all values of t. (c) Find two linearly independent solutions through the degree five term....
View Full Document

This note was uploaded on 11/01/2010 for the course ENGINEERIN PHYS222 taught by Professor Ewqfqw during the Spring '09 term at University of Washington.

Page1 / 6

m238f02t3hs - Mathematics 238 test 3 due Monday note...

This preview shows document pages 1 - 3. Sign up to view the full document.

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