LECTURE 12:
If we were to dive into a deep discussion of how to develop a model for the flow of heat
through a thin, insulated wire whose ends are kept at a constant temperature of 0C and
whose initial temperature distribution is to be specified, we would

Math 39100 K (47062)
Ethan Akin
Office: NAC 6/287
Phone: 650-5136
Email: ethanakin@earthlink.net
Spring, 2016
Contents
First Order Differential Equations, B & D Chapter 2
Linear Growth
Exponential Growth
Existence and Uniqueness Theorem
Separable and Homo

LECTURE 2:
The following things were defined in class:
An nth order linear ODE takes the form
an y(n) + an 1 y(n 1) + + a2 y + a1 y + a0 y = g , where all of the ai and g are
functions of the independent variable. To identify a linear ODE, you should not

LECTURE 7:
An operation L is linear if it has the following properties:
(i)
L(f + g) = Lf + Lg, and
(ii)
L(cf) = cL(f) where c is a constant.
The two methods focused in this lecture are the operator method and the method
of undetermined coefficients. Thes

LECTURE 8:
The method of undetermined coefficients is used for limited types of equations (as we
discussed in class). This method is also only useful for equations only having constant
coefficients. In the discussion here is the method of variation of par

SUPPLEMENTARY NOTES FOR SUMMER 2011
MATH 391
These notes serve as a review of each lecture and include some examples that I may or
may not have examined in lecture. Please take the time to read through these notes,
especially if you find yourself having d

LECTURE 4:
In general, a second order, linear ODE can be written in the form
a ( t ) y + b ( t ) y + c ( t ) y = d ( t )
and, dividing by a(t), which we assume is not zero since we are saying the
equation is second order, then we can write the equation in

LECTURE 3:
f
f
dx + dy .
x
y
The differential form M(x, y)dx + N(x, y)dy is said to be exact (or an exact
differential) if there is a function f (x, y) such that M(x, y) = fx and N(x, y) = fy.
(There is a much more specific mathematical definition of this

LECTURE 5:
The Wronskian of two functions y1(t) and y2(t) is denoted by W(y1, y2)(t) and is
defined as W ( y1 , y2 ) (t) =
y1 (t) y2 (t)
= y1 (t)y2 (t) y1 (t)y2 (t) . We discussed in
y1 (t) y2 (t)
class where the Wronskian comes from and you should be awa

LECTURE 9:
The following notes will help fill in some of the gaps left out of the lecture today which
some of you might find helpful in understanding the theoretical framework of linear
differential equations of higher order. In lecture, I focused on the

LECTURE 13:
I am assuming that boundary value problems are easy problems to digest, so this lecture
focuses on the basics of Fourier series. In the previous lecture, we began the process of
solving a PDE using the assumption that the solution takes on the

LECTURE 10:
A power series about a point x0 is of the form
a (x x )
n=0
n
0
n
where an can be
thought of as a function of a discrete variable n, n = 0, 1, 2, 3, To each number
n is associated a number an, called a coefficient. In our class, the most popul

LECTURE 11:
Integral transforms is personally one of my favorite mathematical topics. The notes
included here are nothing more than an extension of what we have discussed in class
regarding the Laplace transform.
The Laplace transform L cfw_ f ( t ) is de

LECTURE 6:
Here, we discuss the method of reduction of order, which enables us to use one known
solution y1 of a second order homogeneous linear differential equation to find a second
solution y2 such that cfw_y1, y2 form a fundamental solution set. (An a

Math 39100 K (47062)
Ethan Akin
Office: NAC 6/287
Phone: 650-5136
Email: ethanakin@earthlink.net
Spring, 2016
Contents
Second Order Linear Equations, B & D Chapter 4
Second Order Linear Homogeneous Equations with
Constant Coefficients
Polar Form of Comple