MATH 402 - Assignment #1
Due on Friday January 21, 2011
Name —————————————–
Student number —————————
1
Problem 1:
(a) Construct a non-negative smooth function which has its support
inside the interval [0, 1].
(b) Construct a non-negative smooth function which has its support
inside the interval [−2, 2] and it is equal to constant 1 on the interval
[−1, 1].
2
Problem 2:
For each of the following problems, find a function y which satisfies the
associated Euler-Lagrange equation and the given endpoints condition:
2
Z
(y 0 2 + 2yy 0 + y 2 ) dx,
(a)
y(0) = 0,
y(2) = 1.
0
Z
π
2
(b)
(y 0 2 + 2yz + z 0 2 ) dx,
0
Z
(c)
0
π
π
y(0) = 0, y( ) = 1, z(0) = 0, z( ) = 1.
2
2
1
(1 + y 00 2 ) dx,
y(0) = 0, y(1) = 1, y 0 (0) = 1, y 0 (1) = 1.
3
Problem 3:
Show that the Euler-Lagrange equation of the functional
Z b
f (x, y, y 0 , y 00 ) dx
a
has the first integral
∂f
d ∂f
− ( 00 ) = const
0
∂y
dx ∂y
if the integrand does not depend on y, and the first integral
∂f
∂f
d ∂f
0
f −y
− ( 00 ) − y 00 00 = const
0
∂y
dx ∂y
∂y
if the integrand does not depend on x.