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M334Lec21

M334Lec21 - Math 334 Lecture#21 5.2 Power Series Solutions...

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Math 334 Lecture #21 § 5.2: Power Series Solutions, Part I Example. Solve the IVP y + xy + 2 y = 0 , y (0) = 4 , y (0) = - 1 , using a power series y = 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 n =2 n ( n - 1) a n x n - 2 + x n =1 na n x n - 1 + 2 n =0 a n x n = 0 shift index on first sum , multiply x through second sum , multiply 2 through third sum n =0 ( n + 2)( n + 1) a n +2 x n + n =1 na n x n + n =0 2 a n x n = 0 peel off the n = 0 terms from first and third sums n =1 ( n + 2)( n + 1) a n +2 x n + n =1 na n x n + n =1 2 a n x n + 2 a 2 x 0 + 2 a 0 x 0 = 0 combine coefficients of like powers n =1 [( n + 2)( n + 1) a n +2 + na n + 2 a n ] x n + 2( a 2 + a 0 ) = 0 = n =0 0 x n . Two polynomials are the same if and only if the coefficients of like powers are the same: this applies to “infinite” polynomials, or power series.

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M334Lec21 - Math 334 Lecture#21 5.2 Power Series Solutions...

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