Finding n-th Roots
To solve linear differential equations with constant coefcients, we need
to be able to nd the real and complex roots of polynomial equations.
Though a lot of this is done today with calculators and computers, one
still has to know how t
Because of the importance of complex exponentials in differential equa
tions, and in science and engineering generally, we go a little further with
them. Eulers formula denes the exponential to a pure imaginary power.
The denition of
Eulers Formula, Polar Representation
1. The Complex Plane
Complex numbers are represented geometrically by points in the plane:
the number a + i b is represented by the point ( a, b) in Cartesian coordinates.
When the points of the plane are thought of as
Complex Arithmetic Examples
In the following we let z = 2 + 3i and w = 4 + 5i.
1. Real and Imaginary Parts
Re(z) = 2,
Im(z) = 3,
Re(w) = 4,
Im(w) = 5.
Note: the imaginary part does not include i.
2. Addition and Subtraction
z + w = ( 2 + 3i ) + ( 4 + 5i )
Most people think that complex numbers arose from attempts to solve
quadratic equations, but actually they rst appeared in connection with
cubic equations. Everyone knew that certain quadratic equations, like
x2 + 1 = 0
Complex Arithmetic and Exponentials: Introduction
Complex numbers will be a fundamental part of the toolkit for this
course. Using the complex roots of polynomials and allowing exponen
tials with complex exponents will simplify and unify our study of cons
Superposition and the Integrating Factors Solution
1. Another Proof of the Superposition Principle
The superposition principle is so important a concept that it is worth
reviewing yet again. Here we will use the integrating factors formula for
Example: Heat Diffusion
Example. Heat Diffusion
Here we will model heat diffusion with a rst order linear ODE. We will
solve the DE using the method of integrating factors.
Every summer I put my root beer in a cooler, but after a while it still
Solutions to Linear First Order ODEs
1. First Order Linear Equations
In the previous session we learned that a rst order linear inhomogeneous
ODE for the unknown function x = x (t), has the standard form
x + p(t) x = q(t).
(To be precise we should
Solutions by Integrating Factors: Introduction
Our goal in this session is to derive formulas for solving both homo
geneous and inhomogeneous rst order linear ODEs. For the inhomoge
neous equations we will use what are called integrating factors.
1. Superposition Principle for Inputs
We conclude our introduction to rst order linear equations by dis
cussing the superposition principle. This is the most important property of
these equations. In fact, we will see in later sess
Terminology: Systems and Signals
1. Systems, Signals, Input and Output
In the bank example we modeled the amount in a bank account by:
x r x = q ( t ).
Notice that the right-hand side does not depend on x. The left-hand
side represents what happens at t
First order Linear Differential Equations
To start we will dene rst order linear equations by their form. Soon,
we will understand them by their properties. In particular, you should be
on the lookout for the statement of the superposition principle and i
First Order Linear ODEs: Introduction
Linear equations are the most basic and probably the most important
class of differential equations. They will be the main focus of this course.
In this session we will introduce rst order linear ordinary differential
Further Numerical Methods
Eulers method is a rst order method (no relation to rst order equations).
It is possible to show theoretically that for small enough h, the error in
Eulers method is at most C1 h, where C1 is a constant that depends on the
Errors In Eulers Method
As we have seen with the applet, Eulers method is rarely exact. In this
section we try to understand potential sources of error, and nd ways to
estimate or bound it.
1. Common Error Sources
Let us stress this again: the Euler poly
Motivation and Implementation of Eulers Method
1. What Would One Use Numerical Methods For?
The graphical methods described in the previous session give one a
quick feel for how the solutions to a differential equation behave; they can
also be very accura
Numerical Methods: Introduction
The study of differential equations has three main facets:
Analytic methods (also known as exact or symbolic methods).
In the rst two sessions we introduced some of the tools from th
Long-term Behavior: Fences, Funnels and Separatrices
In the video on the Isoclines applet we studied some of the features
of the long-term behavior of the integral curves; among others, the terms
fence, funnel and separatrix were introduced.
Fences and fu
Existence and Uniqueness Theorem for ODEs
The following is a key theorem of the theory of ODEs; it has immediate
(and crucial) consequences for sketching integral curves.
Theorem. Existence and Uniqueness Theorem for ODEs
For any ( a, b) in the region whe
Direction Fields, Isoclines, Integral Curves
Graphical methods are based on the construction of what is called a
direction eld for the equation y = f ( x, y). To get this, we imagine that
through each point ( x, y) of the plane is drawn a little line segm
Geometric Methods: Introduction
In studying the rst order ODE y = f ( x, y) the main emphasis is on
learning different ways of nding explicit solutions. However, you should
realize that most rst order equations cannot be solved explicitly. For such
Separation of Variables
1. Separable Equations
We will now learn our rst technique for solving differential equation.
An equation is called separable when you can use algebra to separate the
two variables, so that each is completely on one side of the equ
Other Basic Examples
Other Basic Examples
Here are some basic examples of DEs taken from math and science.
Except for example 1 we will not give solutions. We will do that and more
with these DEs as we go through the course.
Example 1. (From Calculus)
Introduction: The Most Important DE
1. The Most Important DE
The most important differential equation we will study is
y = a y.
In words the equation says
the rate of change of y is proportional to y.
It is hardly an exaggeration to say that much of
1. Denition of Differential Equations
A differential equation is an equation expressing a relation between a
function and its derivatives. For example, we might know that x is a func
tion of t and
x + 8x + 7x = 0.
Notations for Derivatives
We will write
, y and D y
to all mean the derivative of y with respect to x. Only the rst one species
the independent variable x. In the other two you can only determine the
independent variable from context.
When the indep
Variables and Parameters
Independent and Dependent Variables
When we write a function such as
f ( x ) = 3x2 + 2x + 1
we say that x is an independent variable: it can be freely set to any value
(or any value within the given domain) and the value of the
18.03SC Differential Equations, Fall 2011
Transcript Lecture 6
I a ssume from high school you know how to add and multiply complex numbers
using the relation i^2 = - 1. I'm a little less certain that you remember how to divide
them. I hope you read last n