P45.
2.3 #3.
Proof.
(a). By the strong maximum principle, u(x, t) > 0 in the interior points 0 < x < 1, 0 < t < because the
minimum value of u, 0, is attained at the boundary point, and at the two lateral sides.
(b). We follow the hint. Let (t) = the maxi

P45.
2.3 # 2.
Proof.
(a). Let M(T) = the maximum of u(x, t) in the closed rectangle cfw_0 x l, 0 t T. M(T) is a decreasing
function of T. Indeed, when T1 T2, the rectangle R1 = cfw_0 x l, 0 t T1 is contained in the rectangle
R2 = cfw_0 x l, 0 t T2; the bo

P45.
2.3 # 1.
Proof.
By the maximum principle, u(x, t) = 1 x 2 2kt attains the maximum at the bottom or on the two sides.
When t = 0, 1 x 2 2kt = 1 x 2 attains the maximum at x = 0, i.e., 1 x 2 = 1. When x = 0, 1 x 2 2kt
= 1 2kt attains the maximum 1 in t

P41.
2.2 # 4.
Proof.
The equation utt = uxx has a solution
u(t, x) = f(x + t) + g(x t), 6 where f and g are functions of one variable.
Thus
u(x + h, t + k) = f(x + t + h + k) + g(x t + h k), u(x h, t k)
= f(x + t h k) + g(x t h + k), u(x + k, t + h)
= f(x

P41.
2.2 # 3.
Proof.
(a). The solution u satisfies utt = uxx. For the translate of u = u(x y, t), we differentiate both sides in t
and x,
utt(x y, t) = uxx(x y, t).
(b). Differentiate both sides of utt = uxx in x in any order,
k
k
( x u)tt = ( x u )xx.
Di

P38.
2.1# 11.
Proof. For this equation, we have a particular solution
u(x, t) =
1
16
sin(x + t).
Then we look for solutions to the homogeneous equation
(3x + t)(x + 3t)v = 0.
Then in Lemma 6.2,
a = 3, b = 1, c = 1, d = 3.
Then by (6),
v(x, t) = f(x 3t) +

P24.
1.5 # 2.
Proof.
(a). Fouriers law says that the heat flows from hot to cold regions proportionately to the temperature
gradient. So if the metal rod insulated at the end x = 0, the temperature at x = 0 is
u
x
= 0.
(b). Ficks law says that a chemical

4.3. # 2.
Proof.
(a). We first prove one implication.
Suppose that X(0) = 0. Then X(x) = aX + b. Thus
X (x) = a, 0 x l.
Hence the boundary conditions,
X (0) a0X(0) = 0,
X (l) + alX(l) = 0.
Thus we have
a a0b = 0,
a + al(al + b) = 0.
Hence
(a0 + al)b = a0a