Lecture 13: Series Solutions near Singular Points
March 28, 2007
Here we consider solutions to second-order ODEs using series when the coecients are not
necessarily analytic. A rst-order analogy might be
y +
y
1
C
1
y = 0 =
= = ln |y| = ln |x| = y = .
x
y
Lecture 14: The Laplace Transform
April 11, 2007
Here we take a rst look at the LaPlace1Transform. It should be pointed out that it is one of
many transforms, a term which for our purposes means that it inputs whole functions and outputs
functions as well
Lecture 12: Solutions by Series
Dr. Michael Dougherty
March 26, 2010
Here we look at methods of solving ODEs using series. The basic idea is that we assume the
solution has a power series expansion, and use the equation to nd the coecients of the series.
Lecture 9: LHODEs IV
Dr. Michael M. Dougherty
March 3, 2010
1
The Development So Far
At this point we know that, given an LHODE
an y (n) + an1 y (n1) + + a1 y + a0 = 0,
(1)
with the operator version and characteristic equation given respectively by
an Dn
Lecture 7: LHODEs II
February 26, 2010
1
Characteristic Equation
We saw before that we could take a linear, homogeneous ordinary dierential equation (LHODE)
with constant coecients such as
y 2y 15y = 0,
(1)
rewrite it using dierential operators,
(D3 2D2 1
Lecture 8: LHODEs III
March 1, 2010
In this lecture we will take a rst look at cases where the characteristic equation has complex,
as well as real, solutions. We will delve more deeply into the complex numbers in the next lecture,
and then we will be don
Lecture 11: Variation of Parameters
Dr. Michael M. Dougherty
March 22, 2010
In this lecture we develop a very general method for solving the case of second-order, linear,
nonhomogeneous, constant coecient ODEs for which the right-hand side can not be anni
Lecture 5: Functions Homogeneous of Degree Zero and ODEs
Dr. Michael Dougherty
January 27, 2010
In this lecture we will look at solving a particular type of ODE, which can be written in the form
dy
= f (x, y)
dx
(1)
where f (x, y) is a particular type of
Lecture 1: Differential Equations Introduced
Dr. Michael Dougherty
January 6, 2010
1
First Introduction
Dierential equations are equations which involve derivatives of a function, such as a function y = y(x) (i.e.,
a function we call y or y of x). The sol
Lecture 4: Exact ODEs
Dr. Michael Dougherty
January 22, 2010
Exact equations are rst-order ODEs of a particular form, and whose methods of solutions rely
upon basic facts concerning partial derivatives, exact dierentials and level curves. These are all
Ca
Lecture 6: Linear ODEs with Constant
Coefficients I
Dr. Michael Dougherty
February 12, 2010
1
LHODEs, and Linear Operators Revisited
In this and the next few lectures we will be interested in linear, homogeneous ordinary dierential equations, or LHODEs, w
Lecture 3: First-Order Linear ODEs
Dr. Michael Dougherty
January 13, 2010
1
Some Denitions
Here we briey dene a few terms which will be useful later. In fact, we will revisit the denition of
linear found in Farlows text (p. 12), and put it in a slightly e
Lecture 2: Separable Ordinary Differential
Equations
Dr. Michael Dougherty
January 8, 2010
1
Some Terminology: ODEs, PDEs, IVPs
The dierential equations we have looked at so far are called ordinary dierential equations,
dy d2
or ODEs, because they involve