Chem Differential Eq HW Solutions Fall 2011 33

Chem Differential Eq HW Solutions Fall 2011 33 - Section...

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Section 3.3 Wave Equation, the Method of Separation of Variables 33 where b * 1 = 1 π and all other b * n = 0. The Fourier coefficients of f are b n =2 ± 1 0 x (1 - x ) sin( nπx ) dx. To evaluate this integral, we will use integration by parts to derive ±rst the formula: for a ± =0 , ± x sin( ax ) dx = - x cos( ax ) a + sin( ax ) a 2 + C, and ± x 2 sin( ax ) dx = 2 cos( ax ) a 3 - x 2 cos( ax ) a + 2 x sin( ax ) a 2 + C ; thus ± x (1 - x ) sin( ax ) dx = - 2 cos( ax ) a 3 - x cos( ax ) a + x 2 cos( ax ) a + sin( ax ) a 2 - 2 x sin( ax ) a 2 + C. Applying the formula with a = ,weget ± 1 0 x (1 - x ) sin( nπx ) dx = - 2 cos( nπ x ) ( ) 3 - x cos( nπ x ) + x 2 cos( nπ x ) + sin( nπ x ) ( ) 2 - 2 x sin( nπ x ) ( ) 2 ² ² ² 1 0 = - 2(( - 1) n - 1) ( ) 3 - ( - 1) n + ( - 1) n = - 2(( - 1) n - 1) ( ) 3 = ³ 4 ( ) 3 if n is odd, 0i f n is even. Thus b n = ³ 8 (
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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