Department of Mathematical Sciences Carnegie Mellon University
Basic Examination: Differential Equations February 2021
•This test is closed book: No books, notes, or access to any other relevant materials
(including Internet consultation) are permitted.
•You have 3 hours. The exam has a total of 5 questions and 100 points (20 each).
•You may use without proof standard results from the syllabus (not from homework)
which are independent of the question asked, unless explicitly instructed otherwise.
You must, however, clearly state the result you are using.
Problem 1 : Suppose u:Rn→Ris harmonic in the unit ball B(0,1). For a, c ∈(0,1) let
G(a, c) = Z∂B(0,1)
u(az)u(cz)dS (z).
Show that G(a, c) = G(b, b) where b2=ac. (Hint: Show G(a, b2/a) is independent of a.)
Problem 2 : Suppose U⊂Rnis a bounded domain with smooth boundary. Show that
the initial boundary value problem
ut= ∆u+ZU
u2dx in Ufor t > 0,
u= 0 on ∂U for t > 0,
u(x, 0) = g(x) for x∈U,
may have at most one solution u∈C2(U×[0, T ]), for any T > 0.
Problem 3 : Consider this initial value problem for the inhomogeneous wave equation:
utt −uxx =f(x, t),−∞ <x<∞, t > 0,
u(x, 0) = 0, ut(x, 0) = 0 − ∞ <x<∞,
where fis smooth. Let L > 0 and T > 0, and suppose
f(x, t +T) = f(x, t) for all x∈Rand t > 0, and
f(x, t) = 0 whenever |x|> L, for all t > 0.
(a) Use Duhamel’s principle and d’Alembert’s formula to derive a solution formula for u.
(b) Show that ut+ux= 0 when x=L, for all t > 0.
(c) Show ut(x, t +T) = ut(x, t) for all x≥0, if tis sufficiently large (depending on x).
Problem 4 : Find a solution of
(ux−y)2+u2
y−1=0,
which is smooth and positive in the first quadrant of the unit disk in R2and which vanishes
for y= 0. Show that the (projected) characteristics are circular arcs.
