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**Unformatted text preview: **Math 334 Lecture #21 5.2: Power Series Solutions, Part I Example. Solve the IVP y 00 + xy + 2 y = 0 , y (0) = 4 , y (0) =- 1 , using a power series y = X n =0 a n x n . [This is a mass-spring system with a varying damping constant. The center of the power series guess is the initial time in the initial conditions.] Substitution of the power series guess into the ODE gives X n =2 n ( n- 1) a n x n- 2 + x X n =1 na n x n- 1 + 2 X n =0 a n x n = 0 shift index on first sum , multiply x through second sum , multiply 2 through third sum X n =0 ( n + 2)( n + 1) a n +2 x n + X n =1 na n x n + X n =0 2 a n x n = 0 peel off the n = 0 terms from first and third sums X n =1 ( n + 2)( n + 1) a n +2 x n + X n =1 na n x n + X n =1 2 a n x n + 2 a 2 x + 2 a x = 0 combine coefficients of like powers X n =1 [( n + 2)( n + 1) a n +2 + na n + 2 a n ] x n + 2( a 2 + a ) = 0 = X n =0 x n ....

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