Physics 498/MMA
Handout MT
Oct 28th 2002
Mathematical Methods in Physics I
http://w3.physics.uiuc.edu/
∼
mstone5
Prof. M. Stone
305 Loomis Laboratory
University of Illinois
Do question one, and then as many other questions as you can. Try to answer entire ques
tions. Little, if any, credit will be given for fragmentary answers.
1) Green Function
: Consider the boundary value problem

y
00
=
f
(
x
)
,
y
0
(0) =
y
(1) = 0
.
a) Construct the explicit Green function appropriate to this problem. (5 points)
b) Use your Green function to write down the solution of the boundary value problem as
the sum of two explicit integrals over complementary components of the unit interval.
(5 points)
c) Evaluate the
x
derivative of your solution,
y
(
x
), and confirm that
y
obeys both bound
ary conditions. (5 points)
d) Take one further derivative of your
y
(
x
) and confirm that it does indeed solve the
original problem. (5 points)
2) Bead and string
: A bead of mass
M
is free to slide up and down the
y
axis.
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 Fall '07
 Stone
 Derivative, Boundary value problem, Lagrangian mechanics

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