Math 2280  Lecture 18
Dylan Zwick
Fall 2013
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 equ
MATHUA 262 (Ordinary Differential Equations) Homework 1
Section: 003, Semester: Fall 2016, Instructor: Nicholas Knight
(Revised September 14, 2016)
Due 26Sept. Remember to cite your sources, including your collaborators, and show your work. Please
see t
Lecture 1
Ordinary Differential Equations
MATHUA 262, Section 003, Fall 2016
Nicholas Knight
Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
September 7, 2016
Version: 0.0907
Table of Contents
1
Mathematical Model
Math 2280 Lecture 30

Dylan Zwick
Fall 2013
In todays lecture were going to discuss how to take Laplace trans
forms of step functions, how these relate to translations, and how the cal
culation of Laplace transforms can be simplified for periodic functio
Math 2280  Lecture 28
Dylan Zwick
Fall 2013
Today were going to delve deeper into how to calculate inverse Laplace
transforms. In particular, were going to discuss methods for calculating
inverse Laplace transforms for rational functions, which are funct
Math 2280  Lecture 20
Dylan Zwick
Fall 2013
Today well learn about a method for solving systems of differential
equations, the method of elimination, that is very similar to the elimination
methods we learned about in linear algebra. Well extend this ana
Math 2280  Lecture 21
Dylan Zwick
Fall 2013
Todays lecture will be mostly a review of linear algebra. Please note
that this is intended to be a review, and so the rst 80% of the lecture should
be familiar to you. If its not, please try to review the mate
Math 2280  Lecture 19
Dylan Zwick
Fall 2013
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 x
Math 2280 Lecture 17

Dylan Zwick
Fall 2013
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
resistor, capacitor, and inductor. Well learn that, even though physicall
Math 2280 Lecture 16

Dylan Zwick
Fall 2013
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 tex
Math 2280  Lecture 22
Dylan Zwick
Fall 2013
So far weve examined some systems of rstorder differential equations, and weve learned how to solve those systems by using the method
of elimination. Using this method, we reduced the system to a single
higher
Math 2280  Lecture 23
Dylan Zwick
Fall 2013
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 d
Math 2280  Lecture 24
Dylan Zwick
Fall 2013
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 an
Math 2280  Lecture 25
Dylan Zwick
Fall 2013
So far in our study of how to solve systems of ODEs weve focused
almost exclusively on systems of the form:
x = Ax.
This is just the homogeneous case. Its an extremely important case,
but it can also be viewed
Math 2280  Lecture 27
Dylan Zwick
Fall 2013
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
g
Math 2280  Lecture 26
Dylan Zwick
Fall 2013
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 tra
Math 2280  Lecture 29
Dylan Zwick
Fall 2013
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 w