# APPLIED MATH MASTER’S EXAM January 2011 ```APPLIED MATH MASTER’S EXAM
January 2011
Instructions: Attempt 4 of the following 6 questions. Show all work. Clearly label your
work and attach it to this sheet.
1. Find a two term expansion of the positive singular root X of
x3 − x + = 0
2. Let y(x, ) be the solution of the boundary value problem:
y 00 + y 0 + y 2 = 0 , x ∈ (0, 1)
1
1
, y(1) =
y(0) =
4
2
Find the outer, inner and uniformly valid approximations of y in the limit → 0
assuming a layer exists at x = 0.
3. A functional J : A → IR where
J(y) ≡
Z 1
0
1
xy(x) + y 0 (x)2 dx
2
and
n
A = y ∈ C 2 [0, 1] : y(0) = 1
o
a) Derive the natural boundary condition for y(x).
b) Find the unique extrema ȳ(x) of J over A.
4. Define the operator L and domain D by:
Lu ≡ u00
D ≡ {u ∈ C 2 [0, π] : u(0) = u0 (π) = 0}
a) Find all the eigenvalues λn &gt; 0 and normalized eigenfunctions φn (x) of L
b) Find the series representation g(x, ζ) of the Green’s function solving:
Lu = f (x)
,
u(x) =
Z π
g(x, ζ)dζ
0
5. Find the solution of the integral equation:
Z 1
ex+y u(y) dy + u(x) = e−x
0
6. Use Laplace convolution Theorem to find the bounded general solution of
utt = uxx + g(t) ,
x&gt;0 ,
u(x, 0) = ut (x, 0) = u(0, t) = 0
t&gt;0
```