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Supplementary Problems

u(x, a/2c) = 1 [length of (x − a/2, x + a/2) ∩ (−a, a)]. This takes on different
2c
values for lxl < a/2, for a/2 < x < 3a/2, and for x > 3a/2. Continue in this

manner for each case.

Exercise 126. In Exercise 125, find the greatest displacement, max u(x, t), as
x
a function of t.

Exercise 127. If both φ and ψ are odd functions of x, show that the solution
u(x, t) of the wave equation is also odd in x for all t.

Exercise 128. A spherical wave is a solution of the three-dimensional wave equa-
tion of the form u(r, t), where r is the distance to the origin (the spherical coor-
2
2
dinate). The wave equation takes the form u = c (u + u ) (”spherical wave
r
tt
rr
r
equation”).
2
(a) Change variables v = ru to get the equation for v : v = c v .
rr
tt
(b) Solve for v and thereby solve the spherical wave equation.
(c) Solve it with initial conditions u(r, 0) = φ(r), u (r, 0) = ψ(r), taking both
t
φ(r) and ψ(r) to be even functions of r.
2
x
Exercise 129. Solve u xx − 3u − 4u = 0, u(x, 0) = x , u (x, 0) = e . (Hint:
tt
xt
t
Factor the operator as we did for the wave equation.)
Exercise 130. Solve u xx + u − 20u = 0, u(x, 0) = φ(x), u (x, 0) = ψ(x).
xt
tt
t
Exercise 131. Solve the problem u − u xx = 0, 0 < x < ∞, t > 0,
tt

2
u(0, t) = t , t > 0,

2
u(x, 0) = x , 0 ≤ x < ∞,

u (x, 0) = 6x. 0 ≤ x < ∞.

t
and evaluate u(4, 1) and u(1, 4).
Exercise 132. (a) Solve the following initial boundary value problem for a vi-

brating semi-infinite string which is fixed at x = 0 : u − u xx = 0, 0 < x < ∞,
tt
t > 0,

u(0, t) = 0 t > 0,

u(x, 0) = f(x) 0 ≤ x < ∞,

u (x, 0) = g(x) 0 ≤ x < ∞,

t
2
where f ∈ C ([0, ∞)) and g ∈ C1([0, ∞)) satisfy the compatibility conditions
f(0) = f(0) = g(0) = 0.
Hint: Extend the functions f and g as odd functions f and eg over the real line.
e
Solve the Cauchy problem with initial data f and eg, and show that the restriction
e
of this solution to the half-plane x ≥ 0 is a solution of the problem. Recall that the

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