课程名称︰工程数学下
课程性质︰必修
课程教师︰萧浩明
开课学院：工学院
开课系所︰机械系
考试日期（年月日）︰2020/5/27
考试时限（分钟）：60
是否需发放奖励金：是
试题 :
1. Given the Strurm-Liouville system y'' + λy = 0, y(0) = 0 and y'(π) = 0,
find the eigenvalues and eigenfunctions of the system.
2. In home work (3), we solved the 2-D vibrations of a square drumhead or
(δ^2)z (δ^2)z (δ^2)z
membrane with edges fixed at unit length, ———— = a^2 (————+————)
(δt)^2 (δt)^2 (δt)^2
where a^2 = τ/μ. The drumhead is given an initial transverse displacement
f(x,y) and then released with a velocith g(x,y). Othe boundary conditions
include z(0,y,t) = z(1,y,t) = z(x,0,t) = z(x,1,t) = 0. Find the solution
z(x,y,t) in terms of double Fourier series.
(δ^2)u δu
3. Solve t———— + 2 —— = 2xt and find its general solution.
δxδt δx
4. Solve the Laplace equation (▽^2)u = 0, x > 0, y < 0, u(x,0) = 0, x > 0.
b ,-4 ≦ y ≦ -2
u(0,y) = { |u(x,t)| < M
0 , y < -4 or -2 < y < 0