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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