Page 113 - 6637
P. 113

Solutions to Exercises


                                               (
                         π                         0,    if n is even
                      1 R
               b =          f(θ) sin nθ dθ =        4
                 n
                      π  −π                            , if n is odd
                                                   πn
                                           3             5
                           4    r sin θ    r sin 3θ     r sin 5θ
               u(r, θ) =                +            +             + . . .
                           π       a          3a 3         5a 5

                                      kr
               Exercise 208. [Ae + Be          −kr ]/r.

                                                                     −1
               Exercise 209. u = B + (A − B)(1/a − 1/b) (1/r − 1/b)

                                  1
                                                     2
                                                           2
                                            2
                                      2
               Exercise 211. (r − a ) − [(b − a )/4][(log r. − log a)/(log b − log a)].
                                  4

                                          2
                                    2
               Exercise 212. (r − a )/6 − ab(a + b)(1/a − 1/r)/6

               Exercise 214. (c) γ = 40.


                                        1 2
                                               1 2
               Exercise 215. u = x − y − ax + by + c for any c.
                                        2      2

               Exercise 217. 1 +       3r  sin θ.
                                       a


               Exercise 218.      3r  sin θ − 4  r 3  sin 3θ.
                                   a           a 3


               Exercise 219. 1 +       3a  sin θ.
                                        r


                                            1
               Exercise 221. (b) u = (1 − log r/ log 2) + (           r 2  −  8  ) cos 2θ.
                                            2                         30    15r 2


               Exercise 223. aA log(r/b) + B.

                                                                                  5
               Exercise 225. The first two terms are           2r 2  sin 2θ +  2r 6 πa sin 6θ.
                                                              πa             9

                                                                            √
                                                                                       2
                                                                                 √
               Exercise 227. u(x, y) =        4  P ∞   sin[(2l − 1)y]   sinh(  k+(2l−1) (π−x))  .
                                              π    l=1                 (2l−1) sinh(  k+(2l−1) π)
                                                                                            2

                                             4
               Exercise 231. u(t, θ) = sin θ, or in Cartesian coordinates u(x, y) =                 4y  .
                                                                                                    2
                                             r                                                    x +y 2







                                                              109
   108   109   110   111   112   113   114   115