Homework 1: Vector Spaces
Due by September 2nd
Legendre Polynomials Pn (x) are orthogonal in the interval 1
product
x
1 with respect to scalar
1
dx f(x)g(x).
< f|g >=
(1)
1
You can nd their explicit form in your textbook or online.
1. Normalize Legendre p
Homework 2: Eigenvalues & Eigenvectors
Due by September 9nd
A good example of a linear vector transformation is a rotation by three Euler angles in Euclidean 3D space (Look up Euler angles on the web or in Chapter 3.4 of your textbook). In Cartesian coord
Homework 4: Greens functions
Due by September 28th
A exible rod is xed at its ends at x = 0 and x = 1. Its acted on by a force F(x). If y(x) is the
shape of the rod it must satisfy the following ODE and boundary conditions (you dont have to
show this, jus
Homework 3: Second Order ODEs
Due by September 16th
The differential equation
dy
d2 y
+x
y = (1 x)2
2
dx
dx
has y(x) = x as one of its solutions. Find the most general solution of this ODE.
(1 x)
Find the most general solution of
d2 y
dy
3
+ 2y = sin(x
21.6
Green Functions for First Order Equations
Consider the rst order inhomogeneous equation
L[y] y + p(x)y = f (x)
for x > a
(21.2)
subject to a homogeneous initial condition, B[y] y(a) = 0.
The Green function G(x|) is dened as the solution to
L[G(x|)] =
8
Greens Functions
In this chapter we will investigate the solution of nonhomogeneous dierential
equations using Greens functions. Our goal is to solve the nonhomogeneous
dierential equation
L[u] = f,
where L is a dierential operator. The solution is form
Notes on Greens Functions for Nonhomogeneous Equations
September 29, 2010
The Greens function method is a powerful method for solving nonhomogeneous linear equations Ly(x) =
f (x). The Greens function G(x, a) satises the equation
LG(x, a) = (x a),
where d