The Sheet That Should Be Handed Out
The First Day Of Class
Intermediate Differential Equations
Spring 2012
General Stuff
The Course(s): Intermediate Differential Equations, MA 420 and MA 520
Prerequisites: MA 201 (Calc. C), MA 244 (Intro. to Linear Algebr
36
Systems of Differential Equations:
General Introduction and Basics
Thus far, we have been dealing with individual differential equations. But there are many
applications that lead to sets of differential equations sharing common solutions. In this chap
30
Series Solutions: Preliminaries
(A Brief Review of Innite Series, Power
Series and a Little Complex Variables)
At this point, you should have no problem in solving any differential equation of the form
a
d2 y
dy
+b
+ cy = 0
dx
dx 2
ax 2
or
d2 y
dy
+ bx
32
Power Series Solutions II:
Generalizations and Theory
A major goal in this chapter is to conrm the claims made in theorems 31.1 and 31.3 regarding the
validity of the algebraic method. Along the way, we will also expand both the set of differential
equ
31
Power Series Solutions I: Basic
Computational Methods
When a solution to a differential equation is analytic at a point, then that solution can be represented by a power series about that point. In this and the next chapter, we will discuss when
this c
41
Nonlinear Autonomous Systems of
Differential Equations
We will now attempt to deal with nonlinear systems of differential equations. We will not attempt
to explicitly solve them that is just too difcult. Instead, we will see that certain things we
lear
MA 420/520
4/4/2012
Howell
Homework Set III
At the beginning of class on Friday, April 6, 2012, hand in your work on the following1:
Exercises at the end of Chapt 40 in LN: #1c & 3c (25 pts. total)
Exercises at the end of Chapt 41 in LN: #5a (45 pts.)
Exe
MA 420/520
2/3/2012
Howell
Homework Set I
At the beginning of class on Monday, February 6, 2012, hand in your work on the
following1:
A Brief Review of Elementary Ordinary Differential Equations: Exercise 4a (for 10 pts.)
Exercises at the end of Chapt 30
40
Constant-Matrix Homogeneous Linear
Systems, Part II
40.1
Solutions Corresponding to Complex Eigenvalues
Lets now look at the solutions to x = Ax when the matrix A has complex eigenvalues. Remember, we are assuming the coefcients of A are real-valued co
39
Constant-Matrix Homogeneous Linear
Systems, Part I
39.1
Basics, and Some Fundamental Solutions
We will now further limit ourselves to homogeneous linear systems of the form
dx
= Px
dt
in which the components of the N N matrix P are all real-valued cons
38
General Solutions to Homogeneous
Linear Systems
In this chapter, we will develop the basic linear theory regarding solutions to standard rst-order
homogeneous N N linear systems of differential equations. Fortunately, this theory is very
similar to tha
35
Validating the Method of Frobenius
Let us now focus on verifying the claims made in the big theorems of section 34.1: theorem 34.1
on the indicial equation, and theorem 34.2 on solutions about regular singular points.
We will begin our work in a rather
34
The Big Theorem on the Frobenius
Method, With Applications
At this point, you may have a number of questions, including:
1.
What do we do when the basic method does not yield the necessary linearly independent
pair of solutions?
2.
Are there any shortc
33
Modied Power Series Solutions and the
Basic Method of Frobenius
The partial sums of a power series solution about an ordinary point x0 of a differential equation
provide fairly accurate approximations to the equations solutions at any point x near x0 .
Preparing for the Final
The final is comprehensive. It will be worth 200 points, and will, essentially, be like
taking two tests. There will be roughly
50 to 65 points on power series,
65 to 85 points on linear and nonlinear systems, and
50 to 65 points o
37
First-Order Systems: Basics
In the last chapter, we saw that a certain type of rst-order system in which the rst derivative
of each unknown function is some formula of the collection of unknown functions naturally
arises in a number of applications. We
Chapter & Page: 3718
Systems of Differential Equations: Basics
37.6. (Reformatted for Homework Handout II) A direction eld for
x
y
=
1 2
2 0
x
y
+
0
2
has been sketched to the right below.
Y
3
2
1
X
1
1
2
3
1
Using this system and direction eld:
a. Find a
1
A Brief Review of Elementary Ordinary
Differential Equations
At various points in the material we will be covering, we will need to recall and use material
normally covered in an elementary course on ordinary differential equations. In these notes, we
w
44
Applications to PDE Problems
The use of the Sturm-Liouville theory can be nicely illustrated by solving any of a number of
classical problems. We will consider two: a heat ow problem and a vibrating string problem.
The rst, determining the temperature