MATH 402 - Assignment #5
Due on Monday March 28, 2011
Name —————————————–
Student number —————————
1
Problem 1:
(a) Find all possible piecewise smooth extremals of the functional
Z 4
(y 0 2 − 1)2 dx,
y(0) = 0, y(4) = 2,
0
which have exactly one corner point. Evaluate the integral at these
extremals.
(b) Are there any extremals with more than one corner point?
2
Problem 2:
Find all the piecewise smooth extremals of the functional which yield
an absolute minimum for the integral:
Z 1
y 0 2 (x)(y 0 (x) + 1)2 dx,
y(0) = 0, y(1) = m,
0
where −1 < m < 0.
3
Problem 3:
Discuss all possible piecewise smooth extremals for the functional:
Z b
[2xy(x) + x2 y 0 (x)] dx,
y(a) = a1 , y(b) = b1 .
F (y) =
a
Justify your answer.
4
Problem 4:
Consider the functional
Z 2
4
[(y 0 2 +y 2 )(y−x)2 − y 3 +2xy 2 ] dx,
F (y) =
3
0
where y is a piecewise smooth function.
y(0) = 0,
y(2) = e,
(a) Find the corresponding Euler-Lagrange differential equation and
find out at what points in [0, 2] it must hold.
(b) Show that y = x and y = αex are solutions of the equation found
in part (a).
(c) Find all possible corner points for the extremals of F .
(d) Find a possible extremal for our problem.