Math 2280  Lecture 9
Dylan Zwick
Spring 2014
In the last two lectures weve talked about differential equations for
modeling populations. Today well return to the theme we touched upon
in our second lecture, accelerationvelocity models, and see how diffe
Math 2280  Lecture 43
Dylan Zwick
Spring 2014
Suppose a twodimensional object, or lamina, occupies a region R in
the xyplane. Under some reasonable assumptions we can derive that the
flow of temperature through this object will satisfy the partial diff
Math 2280  Lecture 22
Dylan Zwick
Spring 2014
So far weve examined some systems of firstorder differential equations, and weve learned how to solve those systems by using the method
of elimination. Using this method, we reduce a system to a single highe
Math 2280  Lecture 29
Dylan Zwick
Spring 2014
A few lectures ago we learned that the Laplace transform is linear,
which can enormously simplify the calculation of Laplace transforms for
sums and scalar multiples of functions. The next natural question is
Math 2280  Lecture 33
Dylan Zwick
Spring 2014
Today were going to examine how we find power series solutions to
ordinary differential equations of the form
A(x)y + B(x)y + C(x)y = 0.
Now, as we saw in the second example from our last lecture, were
not al
Math 2280  Lecture 23
Dylan Zwick
Spring 2014
In our last lecture we dealt with solutions to the system:
x = Ax
where A is an n n matrix with n distinct eigenvalues. As promised,
today we will deal with the question of what happens if we have less than
n
Math 2280  Lecture 27
Dylan Zwick
Spring 2014
In our last lecture I introduced the Laplace transform, and we discussed a few of its properties. All nice and good, you may be thinking,
but what does it have to do with solving differential equations? Im so
Math 2280  Lecture 8
Dylan Zwick
Spring 2014
Equilibrium Solutions and Stability
Today were going to talk about the general behavior of autonomous differential equations, and how we can extract information about the behavior
of these differential equatio
Math 2280  Lecture 3: Slope Fields,
Existance, and Uniqueness
Dylan Zwick
Spring 2013
In the last lecture we discussed the firstorder differential equation in
normal form
dy
= f (x, y)
dx
for the special situation where f (x, y) = f (x). There exists no
Math 2280  Lecture 32
Dylan Zwick
Spring 2014
So far in this course weve focused almost exclusively on solving linear differential equations with constant coefficients. But these are, to say
the least, not all the differential equations that are out ther
Math 2280  Lecture 34
Dylan Zwick
Spring 2014
In the last lecture we learned how to solve linear ODEs of the form:
A(x)y + B(x)y + C(x)y = 0,
around ordinary points using power series. Today, well learn how to
solve them around a specific type of singula
Math 2280  Lecture 26
Dylan Zwick
Spring 2014
Today were going to transition from the study of linear dynamical
systems and return to the study of (primarily linear) ordinary differential
equations.
In particular, today well begin talking about Laplace t
Math 2280  Lecture 24
Dylan Zwick
Spring 2014
If we think back to calculus II well remember that one of the most
important things we learned about in the second half of the course were
Taylor series. A Taylor series is a way of expressing a function as a
Math 2280  Lecture 40
Dylan Zwick
Spring 2014
In todays lecture well discuss how Fourier series can be used to solve
a simple, but very important partial differential equation. Namely, the onedimensional heat equation. This is probably the first time you
Math 2280  Lecture 37
Dylan Zwick
Spring 2014
The next major concept about which well learn is that of Fourier series. Fourier series are some of the most interesting and useful objects (or
methods, or whatever) in mathematics. Fourier series are used al
Math 2280  Lecture 42
Dylan Zwick
Spring 2014
Today were going to talk about the wave equation, which is the other
great introductory partial differential equation. The wave equation is a
partial differential equation that models the motion of a plucked
Math 2280  Lecture 1: Differential
Equations  What Are They, Where Do
They Come From, and What Do They
Want?
Dylan Zwick
Spring 2014
Newtons fundamental discovery, the one which he considered necessary to keep secret and published only in the form of an
Math 2280 Lecture 16

Dylan Zwick
Spring 2014
In todays lecture well return to our massspring mechanical system
example, and examine what happens when there is a periodic driving
force f(t) = F
coswt.
0
This lecture corresponds with section 3.6 of the t
Math 2280 Lecture 14

Dylan Zwick
Spring 2014
In todays lecture were going to examine, in detail, a physical system
whose behavior is modeled by a secondorder linear ODE with constant
coefficients. Well examine the different possible solutions, what det
Math 2280  Lecture 18
Dylan Zwick
Spring 2014
Up to this point weve dealt exclusively with initial value problems
where were given the values of a function and some number of its derivatives at a point. So, for example, we may be given the differential e
Math 2280 Lecture 17

Dylan Zwick
Spring 2014
In todays lecture well talk about another very common physical sys
tem that comes up all the time in engineering a closed circuit with a
resistoi capacitor, and inductor. Well learn that, even though physical
Math 2280  Lecture 39
Dylan Zwick
Spring 2014
Today, well develop some of our machinery for using Fourier series,
and see how we can use these Fourier series to solve some simple differential equations.
Todays lecture corresponds with section 9.3 from th
Math 2280  Lecture 41
Dylan Zwick
Spring 2014
Lecture 40 is a long one, so we wont quite be able to get through it in
one day. Consequently, todays lecture is a short one, so that we have time
at the beginning of class to finish lecture 40.
In this lectu
Math 2280  Lecture 2: How To Use
Integration To Solve Differential Equations
Dylan Zwick
Spring 2014
The major topic of calculus II was the study of integrals and how we
integrate functions. Well, it turns out that finding an integral is actually
solving
Math 2280  Lecture 19
Dylan Zwick
Spring 2014
Up to now all the differential equations with which weve dealt have
had one dependent variable and one independent variable. So, a differential equation like:
y + 2xy + 3ex y = sin x,
has independent variable
Math 2280  Lecture 4: Separable
Equations and Applications
Dylan Zwick
Spring 2014
For the last two lectures weve studied firstorder differential equations
in standard form:
y = f (x, y).
We learned how to solve these differential equations for the spec
Math 2280 Lecture 11

Dylan Zwick
Spring 2014
Up to this point weve focused almost exclusively on firstorder differ
ential equations, with only passing references to differential equations of
higher order. Starting today, this will change. hi this lectu
Math 2280  Lecture 7: Population Models
Dylan Zwick
Spring 2014
Today were going to explore one of the major applications of differential equations  population models. Well also explore these models tomorrow in the context of autonomous differential equ
Math 2280  Lecture 6: Substitution
Methods for FirstOrder ODEs and Exact
Equations
Dylan Zwick
Spring 2014
In todays lecture were going to examine another technique that can
be useful for solving firstorder ODEs. Namely, substitutuion. Now, as
with us
Math 2280  Lecture 12
Dylan Zwick
Spring 2014
Today were going to leave the world of secondorder linear ODEs for a
while, and take a look at the wider world of nthorder linear ODEs. In particular, well look at how some of the ideas we talked about last
Math 2280  Lecture 10
Dylan Zwick
Spring 2014
Ive decided not to focus much on computer and numerical methods in
this class, and so consequently we will not be covering sections 2.5 or 2.6 of
the textbook, nor will we have any computer projects. However,
Math 2280001
Week 5, Feb 610, 3.13.4
Mon Feb 6 3.13.2
Finish Friday's notes and examples about second order linear differential equations, including Theorems
1,2,3 and the associated examples, if necessary. The following comments are for us to conside
Math 2280001
Week 10: Mar 2024, sections 5.4, 5.6
questions on homework due tomorrow?
5.4 Massspring systems: untethered massspring trains, and forced oscillation nonhomogeneous
problems.
Consider the massspring system below, with no damping. Althou
Math 2280001
Week 3: Jan 2327, sections 2.12.3
Mon Jan 23
2.1 Improved population models.
Finish any remaining material from Friday.
Let P t be a population at time t . Let's call them "people", although they could be other biological
organisms, decayi
Math 2280001
Week 4: Jan 30Feb 3, sections 2.3, 2.42.6
Mon Jan 30
We'll use last week's notes to discuss section 2.3, improved velocity models
Wed Feb 1
2.42.6 numerical methods for first order differential equations. We will probably meet in LCB 115