ESOE Engineering Mathematics (II) Practice 115.5.12
⚫
Problem Set 12.6
Find the potential in the rectangle 0 ≤ 𝑥 ≤ 20, 0 ≤ 𝑦 ≤ 40 whose upper side is kept at
potential 110 V and whose other sides are grounded.
Answer:
220(1−(−1)𝑛 )
𝑛𝜋𝑥
𝑛𝜋𝑦
𝑢(𝑥, 𝑦) = ∑∞
𝑛=1 𝑛𝜋 sinh(2𝑛𝜋) sin ( 20 ) sinh ( 20 )
⚫ Problem Set 12.7
Using (6), obtain the solution of (1) in integral form satisfying the initial condition
𝑢(𝑥, 0) = 𝑓(𝑥), where𝑓(𝑥) = 𝑥 , 𝑖𝑓|𝑥| < 1 𝑎𝑛𝑑 0 otherwise
Answer:
∞ 2
𝑢(𝑥, 𝑡) = ∫0
𝜋𝑝2
2 2
(sin 𝑝 − 𝑝 cos 𝑝) sin(𝑝𝑥) 𝑒−𝑐 𝑝 𝑡 𝑑𝑝