Chem Differential Eq HW Solutions Fall 2011 68

Chem Differential Eq HW Solutions Fall 2011 68 - 68 Chapter...

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68 Chapter 4 Partial Differential Equations in Polar and Cylindrical Coordinates Solutions to Exercises 4.7 1. Bessel equation of order 3. Using (7), the Frst series solution is J 3 ( x )= ± k =0 ( - 1) k k !( k + 3)! ² x 2 ³ 2 k +3 = 1 1 · 6 x 3 8 - 1 1 · 24 x 5 32 + 1 2 · 120 x 7 128 + ··· . 5. Bessel equation of order 3 2 . The general solution is y ( x c 1 J 3 2 + c 2 J - 3 2 = c 1 ´ 1 1 · Γ( 5 2 ) ² x 2 ³ 3 2 - 1 1 · Γ( 7 2 ) ² x 2 ³ 7 2 + µ + c 2 ´ 1 1 · Γ( - 1 2 ) ² x 2 ³ - 3 2 - 1 1 · Γ( 1 2 ) ² x 2 ³ 1 2 + µ . Using the basic property of the gamma function and (15), we have Γ( 5 2 3 2 Γ( 3 2 3 2 1 2 Γ( 1 2 3 4 π Γ( 7 2 5 2 Γ( 5 2 15 8 π - 1 2 Γ( - 1 2 )=Γ( 1 2 π Γ( - 1 2 - 2 π. So y ( x c 1 2 πx ´ 4 3 x 2 4 - 8 15 x 4 16 + µ c 2 2 ( - 1) ´ - 1 2 2 x - x 2 -··· µ = c 1 2 ´ x 2 3 - x 4 30 + µ + c 2 2 ´ 1 x + x 2 µ 9. Divide the equation through by x 2 and put it in the form y ±± + 1 x y ± + x 2 - 9 x 2 y = 0 for x> 0 . Now refer to Appendix A.6 for terminology and for the method of ±robenius that we are about to use in this exercise. Let p ( x 1 x for q ( x x 2 - 9 x 2 . The point x = 0 is a singular point of the equation. But since xp ( x ) = 1 and x 2 q ( x x 2 - 9 have power series expansions about 0 (in fact, they are already given by their power series expansions), it follows that
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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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