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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 first the formula: for a = 0, x sin( ax ) dx = - x cos( a x ) a + sin( a x ) a 2 + C, and x 2 sin( ax ) dx = 2 cos( a x ) a 3 - x 2 cos( a x ) a + 2 x sin( a x ) a 2 + C ; thus x (1 - x ) sin( ax ) dx = - 2 cos( a x ) a 3 - x cos( a x ) a + x 2 cos( a x ) a + sin( a x ) a 2 - 2 x sin( a x ) a 2 + C. Applying the formula with a = , we get 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 1 0 = - 2 (( - 1) n - 1) ( ) 3 - ( - 1) n + ( - 1) n = - 2 (( - 1) n - 1) ( ) 3 = 4 ( ) 3 if n is odd, 0 if n is even. Thus
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