ENGINEERING MATHEMATICS II
Homework 3
Instructor: Hao-Ming Hsiao
∂2 u
∂u
1. Solve t ∂x ∂t + 2 ∂x = x 2 and find its general solution.
2. Solve the vibrations of a string
∂2 y
∂t2
(15%)
∂2 y
= 16 ∂x2 , 0 < x < 2, t > 0, subjected to the
conditions y(0, t) = y(2, t) = 0, y(x, 0) = 6sinx – 3sin4x. y(x, t) is bounded and
yt(x, 0) = 0 where yt is the differentiation with respect to t.
(20%)
3. Extend Problem (2) to 2-D vibrations of a square drumhead or membrane with
edges fixed at unit length,
∂2 z
∂t2
∂2 z
∂2 z
= a2 (∂x2 + ∂y2 ) where a2 = / The drumhead is
given an initial transverse displacement and then released.
(a) assuming the initial transverse displacement z is f(x, y) and the initial velocity
zt is zero, express all conditions related to this boundary value problem,
(b) find its solution z(x, y, t) in terms of double Fourier sine series,
(c) the cosine part in the solution z(x, y, t) can also be expressed as cos(2fmnt),
where fmn is called the natural modes of vibration. It is known that, in order to
create music, the higher modes must be integer multiples of the lowest
frequency. Determine whether this square drumhead could create music tone
or not.
(20%)
4. Find a bounded solution to Laplace’s equation ∇2 v = 0 for the half plane y > 0 if
v takes on the value f(x) on the x-axis v(x, 0) = f(x).
(20%)
5. Solve the PDE
∂u
∂t
∂2 u
− ∂x2 + u = 5 sin2x + 4x, 0 < x < , t > 0, subjected to the
conditions u(0, t) = 0, u(, t) = 4
(25%)
Hint: Assume u(x, t) = w(x, t) + f(sin2x) + g(x) and transfer the PDE from u to w.