MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 7
Problem 1, p. 141142
Use the WKB method to nd an approximate solution to the initial value
problem
d2 y
2
1 + x2 y = 0 ,
1,
2
dx
dy
y (0) = 0 ,
(0) = 1 .
dx
(1)
Solution: To bring the equ
1
MATH 4525, Applied Analysis, Spring 2013
Solutions to Homework 4
Problem 2, p. 100104
Consider the initial value problem
d2 u
u = tu,
dt2
u(0) = 1 ,
t > 0,
(1)
du
(0) = 1 .
dt
Find a two-term perturbative approximation for 0 < 1, as well as a six-term
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 1
Problem 1, p. 78
A pendulum executing small vibrations has period P and length l, and m is the
mass of the bob. Can P depend only on l and m? If we assume P depends on l
and the accelerati
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 5
Problem 8c, p. 100104
Use the Poincar-Lindstedt method to obtain a two-term approximation to the
e
following equation:
d2 y
dy 2
+y= y 1
,
dt2
dt
(1)
dy
y (0) = 1 ,
(0) = 0 .
dt
Solution:
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 3
Problem 7, p. 229230
Find a power series solution to the dierential equation
(2x 1) y + 2y = 0 .
Determine the radius of convergence of the resulting series, and identify the
series soluti
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 2
Problem 1, p. 3035
Let u(t), 0 t b, be a given smooth function. If
M = max |u(t)| ,
then u can be scaled by M to obtain the dimensionless dependent variable
U = u/M . A timescale can be ta
1
MATH 4525, Applied Analysis, Spring 2013
Solutions to Homework 10
Problem 5, p. 184185
Find the extremals for the following functionals.
(b.)
3
J [y ] =
0
e2 x (y )2 y 2 dx ,
y (0) = 1 ,
y (3) free .
(c.)
1
J [y ] =
0
1 2
(y ) + y y + y + y dx ,
2
1
y (
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 9
Problem 2, p. 158
Consider the functional
1
J [y ] =
(1 + x)
0
2
dy
dx
dx ,
where y (x) is twice continuously dierentiable and satises
y (0) = 0 ,
y (1) = 1 .
Of all functions of the form
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 8
Problem 4, p. 148150
Verify the following approximations for large
(a.)
e t ln 1 + t2 dt
0
1.
2!
1 4!
+ .
3
2 5
(1)
(b.)
1
e + .
2
2
1 + t e (2t t ) dt
0
(2)
Solution: To compute the as
MATH 4525, Applied Analysis, Spring 2013
1
Solutions to Homework 6
Problem 1, p. 121123
Use singular perturbation methods to obtain a leading-order, uniformly-valid
approximate solution to each of the following boundary-value problems. In each
case, assum