1. Every first-order PDE involving u(x, y) can be expressed in the form
F (x, y, u, ux , uy ) = 0
for some function F . The associated characteristic equations are given by
 ′

x (s) = Fp






′


 y (s) = Fq

′
u (s) = pFp + qFq
,


′


p (s) = −Fx − pFu 



 ′

q (s) = −Fy − qFu
where p = ux and q = uy as usual. You may use these equations without proof.
(a) [6 points] Solve the initial value problem
xux (x, y) + uy (x, y) = u(x, y),
u(x, 0) = f (x).
(b) [6 points] Does the following initial value problem have any solutions?
ux (x, y) + y 2 uy (x, y) = u(x, y),
u(x, 0) = ex .
(c) [8 points] Solve the initial value problem
ux (x, y)2 + uy (x, y) = y,
u(x, 0) = x.
2. Suppose that A is a bounded, open subset of R2 and let c : A → R be a function such
that c(x, y) ≤ 0 for all (x, y) ∈ A.
(a) [10 points] Prove the following version of the extended maximum principle. If
uxx (x, y) + uyy (x, y) + c(x, y)u(x, y) ≥ 0
for all (x, y) ∈ A
,
u(x, y) ≤ 0
for all (x, y) ∈ ∂A
then one actually has u(x, y) ≤ 0 for all (x, y) ∈ A ∪ ∂A.
(b) [6 points] Suppose that both u(x, y) and v(x, y) satisfy the nonlinear equation
uxx (x, y) + uyy (x, y) = u(x, y)3 for all (x, y) ∈ A.
Assuming that u ≤ v on the boundary ∂A, show that u ≤ v within A ∪ ∂A.
(c) [4 points] Show that the result of part (a) does not necessarily hold in the case
that the function c is positive at all points.
3. (a) [6 points] Suppose that y(x) is an extremal of
J(y) =
Z
b
L(x, y(x), y ′ (x)) dx.
a
Show that y(x) must satisfy the Euler-Lagrange equation
d
Ly ′ = Ly
dx
subject to the natural boundary conditions Ly′ (x, y(x), y ′ (x)) = 0 when x = a, b.
(b) [8 points] Find the critical points of
J(y) =
Z
0
π/3 ′
2
′
2
y (x) + 4y(x)y (x) − y(x) + 6y(x) dx.
(c) [6 points] Let J(y) be the functional of the previous part and suppose y(x) is a
critical point of J(y). Show that J(y) attains a local minimum at y(x).