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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....
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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.
- Spring '09