Assignment 8
Math 22B
Due June 3. 2015
For each of the following systems of equations:
(i) Rewrite in matrix form.
(ii) Find the eigenvalues and eigenvectors of the matrix.
(iii) Find the general solution which is real valued.
(iv) Calculate the Wronskian

Math 22B
Comments on January 14, 2013, lecture
In deriving the small 0 expansion of the period of the pendulum T , the
following expansion was used:
cos cos 0 =
12
14
2
4
( 0 )
0 +
2!
4!
Here is a better explanation of this than given in the lectures:

L1.3(#1)
1)
EDU> help format
FORMAT Set output format.
FORMAT with no inputs sets the output format to the default appropriate
for the class of the variable. For float variables, the default is
FORMAT SHORT.
FORMAT does not affect how MATLAB computations

HW #6 Solutions1
5.5.6:
We are again asked to analyze the DE
dx
= Ax
dt
where now A is a 2 2 matrix given by (5.16). The eigenvalues of A are the
roots of the characteristic polynomial of A. A simple calculation shows we must
nd the roots to
2 4 + 3 + 2 =

HW #1 Solutions1
1.3.2:
We are asked to linearize the DE
g
+ sin = 0
near the point = . Let = then = and
sin() = sin( + ) = sin() .
Thus the linearized DE is
g
= 0.
The dierence between this linearization and the one at = 0 is the sign in front
of . We

HW #9 Solutions1
7.5.2:
First recall the denition of the inner product
(, ) =
(x)(x) dx
where the bar denotes complex conjugation.2 Relative to this inner product the
operator x (multiplication by x) and the momentum operator
p = i
d
dx
are self-adjoint (

HW #8 Solutions1
6.9.7 We are asked to solve the two-dimensional Helmholtz equation
U + k 2 U = 0
in the rectangular domain 0 x a, 0 y b subject to the boundary
conditions that U (x, y) vanish on the boundary of the rectangle. We write
U (x, y) = X(x)Y (y

HW #7 Solutions1
6.9.3 We begin with (6.34) of the notes
1
2
2
(1 + k1 )1 + k1 2
2
k2 1 (2 + k2 )2 .
=
=
Dene
1
2
=
.
Then the above DEs are in matrix form
= A
where
A=
(1)
2
(1 + k1 )
k1
2
k2
(2 + k2 )
We look for a solution of (1) of the form
(t) = eit

Practice Midterm
Math 22B
1. For the following dierential equations, write down the order, determine whether they are linear
or nonlinear, and homogeneous or non-homogeneous.
3
y
(a) d 3 + ( dy )2 = 0
dt
dt
(b) dy + 2y 3 = e4t
dt
(c) dy + t3 y = 0
dt
2. S

Final Review
Math 22B
For each dierential equation, write its order, show it is linear or non-linear, and if it
is linear, write whether it is homogeneous or non-homogeneous. For review, you should
also review Fourier series, systems of dierential equatio

Assignment 5
Math 22B
Due May 1, 2015
1. (a) Show y1 (x) = cos(kx) and y2 (x) = sin(kx) are solutions to y + k 2 y = 0, where k = 0.
(b) Find the Wronskian for this pair of solutions. For what initial conditions can you nd a solution.
2. Find a pair of fu

Assignment 8
Math 22B
Due June 3, 2015
For each of the following systems of equations:
(i) Rewrite in matrix form.
(ii) Find the eigenvalues and eigenvectors of the matrix.
(iii) Find the general solution which is real valued.
(iv) Calculate the Wronskian

Assignment 7
Math 22B
Due May 22, 2015
1.Solve the following dierential equations using the Laplace Transform.
(a) y + 3y + 2y = 0, y(0) = 1, y (0) = 0
(b) y + 2 y = cos(2t), omega2 = 4, y(0) = 1, y (0) = 0
(c) y (4) 4y + 6y 4y + y = 0, y(0) = 0, y (0) =

Assignment 6
Math 22B
Due May 15, 2015
1. Use the method of undetermined coecients to nd the general solutions to the following dierential equations.
(a) y + 2y = 3 + 4 sin(2t)
(b) 2y + 3y + y = t2 + 3 sin(t)
(c) y + y + 4y = 2 sinh(t)
2. Use the variatio