SM212P Test 4, 4-28-2006
Prof Joyner
Name: ANSWERS
USNA math tables, TI92s okay. Closed books, closed notes. No discussion
of this exam until after 6th period. Work individually. Show all work.
1. Solve
x = 4y, x(0) = 100,
y = x, y (0) = 25,
using the eig

Eigenvalue method for systems of DEs
Prof. Joyner, 10-24-20071
In this section, we will try to solve the
PROBLEM: Solve
x = ax + by,
y = cx + dy,
Let
A=
x(0) = x0 ,
y (0) = y0 .
ab
cd
In matrix notation, the system of DEs becomes
x = Ax, x(0) =
where x =

Fourier series, sine series, cosine series
Prof. Joyner1
History: Fourier series were discovered by J. Fourier, a Frenchman who
was a mathematician among other things. In fact, Fourier was Napoleons
scientic advisor during Frances invasion of Egypt in the

An introduction to systems of DEs
Lanchesters equations for battle
Prof. Joyner1
A goal of military analysis is a means of reliably predicting the outcome
of military encounters, given some basic information about the forces status.
The case of two combat

Solving ODEs:
using the power series method, I
Prof. Joyner1
In this part, we recall some basic facts about power series and Taylor
series. We will turn to solving DEs in part II.
Roughly speaking, power series are simply innite degree polynomials
f (x) =

x
U p
t gfU
t U lp l`P x U T sr
X t
U p
t g f U x U f P U SQ P
x
gU dt VP x dgd y rTB t U rp gP x U T srgU dt dERVTR`H
P U g U t
X t
V8 l| t Rs' T p !d dt
gfU
g Pf t f x
x P P
!t R sRR| t P !TRP d T be0gR)VR| t P dt id'cfw_ lX gA
p U U U
Q U t

Introduction to DEs
Prof. Joyner, 8-15-20071
But there is another reason for the high repute of mathematics: it is mathematics that oers the exact natural sciences
a certain measure of security which, without mathematics, they
could not attain.
- Albert E

Undetermined coecients in constant coecient ODEs
Prof. Joyner1
The method of undetermined coecients [U] can be used to solve the
following type of problem.
PROBLEM: Solve
ay + by + cy = f (x),
(1)
where a = 0, b and c are constants. (Even the case a = 0 c

Second order ODEs - variation of parameters
Prof. Joyner, 9-3-20071
Consider an ordinary constant coecient non-homogeneous 2nd order
linear dierential equation,
ay + by + cy = F (x)
where F (x) is a given function and a, b, and c are constants. (For the m