Chem Differential Eq HW Solutions Fall 2011 95

# Chem Differential Eq HW Solutions Fall 2011 95 - Section...

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Section 6.2 Sturm-Liouville Theory 95 For the third integral, we have ± 1 - 1 g ( x ) h ( x ) w ( x ) dx = ± 1 - 1 2 x ( - 1+4 x 2 ) ² 1 - x 2 dx =0 , because we are integrating an odd function over a symmetric interval. 9. In order for the functions 1 and a + bx + x 2 to be orthogonal, we must have ± 1 - 1 1 · ( a + bx + x 2 ) dx Evaluating the integral, we ±nd ax + b 2 x 2 + 1 3 x 3 ³ ³ ³ 1 - 1 =2 a + 2 3 a = - 1 3 . In order for the functions x and 1 3 + bx + x 2 to be orthogonal, we must have ± 1 - 1 1 · ( 1 3 + bx + x 2 ) xdx Evaluating the integral, we ±nd 1 6 x 2 + b 3 x 3 + 1 4 x 4 ³ ³ ³ 1 - 1 = b 3 b . 13. Using Theorem 1, Section 5.6, we ±nd the norm of P n ( x )tobe ± P n ± = ´± 1 - 1 P n ( x ) 2 dx µ 1 2 = ´ 2 2 n +1 µ 1 2 = 2 2 n . Thus the orthonormal set of functions obtained from the Legendre polynomials is 2 2 n P n ( x ) ,n , 2 ,.... 17. For Legendre series expansions, the inner product is de±ned in terms of
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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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