In the spherical coordinates , , ( > 0, 0 < < 2 , 0 < < ) defined by the equations x =

Chapter 11, Problem 9

(choose chapter or problem)

In the spherical coordinates , , ( > 0, 0 < < 2 , 0 < < ) defined by the equations x = cos sin , y = sin sin , z = cos , Laplaces equation is (see in Section 10.8) 2u + 2u + (csc2 )u + u + (cot )u = 0. a. Show that if u(, , ) = P()( )(), then P,, and satisfy ordinary differential equations of the form 2P__ + 2 P_ 2P = 0, > 0, __ + 2 = 0, 0 < < 2, (sin2 )__ + (sin cos )_ +(2 sin2 2) = 0, 0 < < . The differential equation for P is of the Euler type, while the one for is related to Legendres equation. b. Show that if u(, , ) is independent of , then the first equation in part (a) is unchanged, the second is omitted, and the third becomes (sin2 )__ + (sin cos )_ + (2 sin2 ) = 0. c. Define a new independent variable s = cos . Then show that the equation for in part b becomes (1 s2) d2 ds2 2s d ds + 2 = 0, 1 s 1. Note that this is Legendres equation.