1. Verify that a solution to the heat equation (1.1) on the domain −∞ < x <
∞, t > 0 is given by
u(x, t) = 1√ 4πkt
e−x2/4kt.
For a fixed time, the reader should recognize this solution as a bell-shaped
curve. (a) Pick k = 0.5. Use software to sketch several time snapshots on
the same set of coordinate axes to show how the temperature profile evolves
in time. (b) What do the temperature profiles look like as t → 0? (c) Sketch
the solution surface u = u(x, t) in a domain −2 ≤ x ≤ 2, 0.1 < t < 4. (d)
How does changing the parameter k affect the solution?
2. Verify that u(x, y) = ln √
x2 + y2 satisfies the Laplace equation
uxx + uyy = 0
for all (x, y) �= (0, 0).
3. Find the general solution of the equation uxy(x, y) = 0 in terms of two
arbitrary functions.
4. Derive the solution u = u(x, y) = axy + bx+ cy + d (a, b, c, d constants),
of the PDE
u2 xx + u2
yy = 0.
Observe that the solution does not explicitly contain arbitrary functions.
10 1. The Physical Origins of Partial Differential Equations
5. Find a function u = u(x, t) that satisfies the PDE
uxx = 0, 0 < x < 1, t > 0,
subject to the boundary conditions
u(0, t) = t2, u(1, t) = 1, t > 0.
6. Verify that
u(x, t) = 1
2c
∫ x+ct
x−ct
g(s)ds
is a solution to the wave equation utt = c2uxx, where c is a constant and g is
a given continuously differentiable function. Hint: Here you will need to use
Leibniz’s rule for differentiating an integral with respect to a parameter
that occurs in the limits of integration:
d
dt
∫ b(t)
a(t)
F (s)ds = F (b(t))b′(t)− F (a(t))a′(t).
7. For what values of a and b is the function u(x, t) = eat sin bx a solution to
the heat equation
ut = kuxx.
8. Find the general solution to the equation uxt + 3ux = 1. Hint: Let v = ux
and solve the resulting equation for v; then find u.
9. Show that the nonlinear equation ut = u2 x+uxx can be reduced to the heat
equation (1.1) by changing the dependent variable to w = eu.
10. Show that the function u(x, y) = arctan(y/x) satisfies the two-dimensional
Laplace’s equation uxx + uyy = 0.
11. Show that e−ξy sin(ξx), x ∈ R, y > 0, is a solution to uxx + uyy = 0 for
any value of the parameter ξ. Deduce that
u(x, y) =
∫ ∞
0
c(ξ)e−ξy sin(ξx)dξ
is a solution to the same equation for any function c(ξ) that is bounded
and continuous on [0,∞). Hint: The hypotheses on c allow you to bring
a derivative under the integral sign. [This exercise shows that taking inte-
grals of solutions sometimes gives another solution; integration is a way of
superimposing, or adding, a continuum of solutions.]
