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Solutions to Exercises

Exercise 100. In view of the diffusion equation we obtain (a) λ = −µ =
[ω/(2k)] 1/2 , where k is the diffusion constant; (b) x = (365) 1/2 x ; (c) only the
d
a
annual variation is significant at this depth and has a phase µ x = ln 20 behind
a a
the surface. Thus the coldest day is 1 February +(365 ln 20)/(2π) days ≈ 23 July.

Exercise 101. In view of physical considerations it is easy to prove
(a) Parabolic, open, Dirichlet u(x, 0) given, Neumann ∂u/∂x = 0 at x = ±L/2

for all t;
(b) elliptic, closed, Dirichlet;
(c) elliptic, closed, Neumann ∂u/∂n = σ/ε ;
0
(d) hyperbolic, open, Cauchy.

Exercise 104. Try a simple polynomial in x and t.

00
Exercise 108. An equilibrium temperature distribution has the form (a) u = 0 in
1
00
0 ≤ x ≤ L , u = 0 in L ≤ x ≤ L + L , together with four boundary and jump
1
1
1
2
2
conditions. Hence u (x) = ax + b and u (x) = cx + d. Solve for a and b using the
1
2
four conditions. (b) u = 10x/7 and u = 10(2x − 3)/7.
1
2

Exercise 109. Solve the ODE. The solution is unique only if L is not an integer
multiple of π.

Exercise 110. In view of multipoint conditions we obtain (a) Take the difference
of two solutions and solve the ODE. Not unique. (b) Integrate the equation from
0 to l. The function f(x) must have zero average.

1
x
Exercise 122. e cosh ct + sin x sin ct.
c

p
Exercise 124. (l/4 − a) ρ/T.

Exercise 126. Let m(t) = max u(x, t). Then m(t) = t for 0 ≤ t ≤ a/c, and
x
m(t) = a/c for t ≥ a/c.

Exercise 128. Our solution of the spherical wave equation has the form
1
(b) u(r, t) = [f(r + ct) + g(r − ct)].
r R
(c) 1 {(r + ct)φ(r + ct) + (r − ct)φ(r − ct)} + 1 r+ct sφ(s)ds.
2r 2cr r−ct

105