Math2930_WS8_Solutions

Math2930_WS8_Solutions - Timothy Eng [email protected] Dis...

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Unformatted text preview: Timothy Eng [email protected] Dis 202,210,212 Week 10: Chapters 3.8,8.1,8.2 Math 2930: Discussion Worksheet 8 On Radius of Convergence: When facing c n +1 = ...c n , you can make use the equation by simply dividing by c n and to evaluating the limit using the following equation. ρ = lim n →∞ | c n c n +1 | but when facing recurrence relations with higher than first order, for example c n +2 = ...c n +1 + ...c n , you will be required to use Theorem 1 on p.519. 1) Solve y 00 + y- 2 y = 0; y (0) = 1 , y (0) =- 2 ∞ X n =2 n ( n- 1) c n x n- 2 + ∞ X n =1 nc n x n- 1- 2 ∞ X n =0 c n x n = 0 Shifting summations to start at n = 0 ∞ X n =0 ( n + 2)( n + 1) c n +2 x n + ∞ X n =0 ( n + 1) c n +1 x n- 2 ∞ X n =0 c n x n = 0 Because 0 = ∑ ∞ n =0 0: ( n + 2)( n + 1) c n +2 + ( n + 1) c n +1- 2 c n = 0 Solving for recurrence relation, the relation between new and old coefficients: c n +2 = 2 c n- ( n + 1) c n +1 ( n + 2)( n + 1) ← To solve the equation, we know the solution is in the form of an infinite series of polynomials with unknown...
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Math2930_WS8_Solutions - Timothy Eng [email protected] Dis...

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