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Supplementary Problems

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Exercise 168. Solve u = c u +k for 0 < x < l, with the boundary conditions
xx
tt
u(0, t) = 0, u (l, t) = 0 and the initial conditions u(x, 0) = 0, u (x, 0) = V. Here
x
t
k and V are constants.
t
2
Exercise 169. Solve u tt = c u xx + e sin 5x for 0 < x < π with u(0, t) =
u(π, t) = 0 and the initial conditions u(x, 0) = 0, u (x, 0) = sin 3x.
t
2
Exercise 170. Solve u = c u xx + g(x) sin ωt for 0 < x < l, with u = 0 at both
tt
ends and u = u = 0 when t = 0. For which values of ω can resonance occur?
t
(Resonance means growth in time.)

2
Exercise 171. Repeat Exercise 170 for the damped wave equation u = c u −
tt
xx
ru + g(x) sin ωt, where r is a positive constant.
t
Exercise 172. Solve u = ku xx in (0, l), with u(0, t) = 0, u(l, t) = At, u(x, 0) =
t
0, where A is a constant.

Exercise 173. The wave equation describing the transverse vibrations of a stret-
2
ched membrane under tension T and having a uniform surface density ρ is T( ∂ u +
∂x 2
2
2
∂ u ) = ρ ∂ u . Find a separable solution appropriate to a membrane stretched on a
∂y 2 ∂t 2
frame of length a and width b, showing that the natural angular frequencies of such
2
2
a membrane are ω = π T n 2 + m 2 ), where n and m are any positive integers.
(
ρ a 2 b 2
Solution: u(x, y, t) = sin(πnx/a) sin(πmy/b)(A sin ωt + b cos ωt).
Exercise 174. Using the separation of variables method find a (formal) solution

of a vibrating string with fixed ends:
2
u − c u xx = 0, 0 < x < L, 0 < t,
tt
u(0, t) = u(L, t) = 0 t ≥ 0,
u(x, 0) = f(x) 0 ≤ x ≤ L,
u (x, 0) = g(x) 0 ≤ x ≤ L.
t

Exercise 175. Find a formal solution of the problem
u = u xx 0 < x < π, t > 0,
tt
u(0, t) = u(π, t) = 0 t ≥ 0,
3
u(x, 0) = sin x 0 ≤ x ≤ π,
u (x, 0) = sin 2x 0 ≤ x ≤ π.
t
Exercise 176. Consider the following problem:
u − u xx = 0 0 < x < 1, t > 0,
tt
u (0, t) = u (1, t) = 0 t ≥ 0,
x
x
u(x, 0) = f(x) 0 ≤ x ≤ 1.
2
Solve the problem with f(x) = 2 sin 2πx.

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