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Chem Differential Eq HW Solutions Fall 2011 74

Chem Differential Eq HW Solutions Fall 2011 74 - 74 Chapter...

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74 Chapter 4 Partial Differential Equations in Polar and Cylindrical Coordinates Solutions to Exercises 4.8 1. (a) Using the series definition of the Bessel function, (7), Section 4.7, we have d dx [ x - p J p ( x )] = d dx k =0 ( - 1) k 2 p k !Γ( k + p + 1) x 2 2 k = k =0 ( - 1) k 2 p k !Γ( k + p + 1) d dx x 2 2 k = k =0 ( - 1) k 2 k 2 p k !Γ( k + p + 1) 1 2 x 2 2 k - 1 = k =0 ( - 1) k 2 p ( k - 1)!Γ( k + p + 1) x 2 2 k - 1 = - m =0 ( - 1) m 2 p m !Γ( m + p + 2) x 2 2 m +1 (set m = k - 1) = - x - p m =0 ( - 1) m m !Γ( m + p + 2) x 2 2 m + p +1 = - x - p J p +1 ( x ) . To prove (7), use (1): d dx [ x p J p ( x )] = x p J p - 1 ( x ) x p J p - 1 ( x ) dx = x p J p ( x ) + C. Now replace p by p + 1 and get x p +1 J p ( x ) dx = x p +1 J p +1 ( x ) + C, which is (7). Similarly, starting with (2), d dx [ x - p J p ( x )] = - x - p J p +1 ( x ) - x - p J p +1 ( x ) dx = x - p J p ( x ) + C x - p J p +1 ( x ) dx = - x - p J p ( x ) + C. Now replace p by p - 1 and get x - p +1 J p ( x ) dx = - x - p +1 J p - 1 ( x ) + C, which is (8). (b) To prove (4), carry out the differentiation in (2) to obtain x - p J p ( x ) - px - p - 1 J p ( x ) = - x - p J p +1 ( x )
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