Simplifying A sin( x) + B cos( x)
A and B are any two real non-zero numbers and > 0.
We want to write E = A sin ( x) + B cos ( x) = R sin ( x + ) , for suitable R > 0 and .
R is called the amplitude and is called the phase (or phase shift) of the expressi
PARTIAL FRACTIONS
Given a rational function, i.e., one of the form:
N ( x)
where N ( x) and D ( x) are polynomials, first
D ( x)
perform a long division (if necessary), so that we may assume that the degree of N ( x) is less than the degree
N ( x)
of D (
The Phase Line, Stability and Inflection Points - Answers to the Exercises
1: Logistic Growth: p = p(a b p) , where a and b are positive constants.
The equilibrium points are p = 0 and
p = 0 is unstable (a source) and p =
a
.
b
a
is stable (a sink).
b
a
The Wronskian of a Set of Three Functions
The Wronskian of three functions is given by:
y1
y2
y3
W(y1, y2 , y3 ; t) = y1
y2
y3 .
y1
y 2
y3
For the Cauchy-Euler differential equation: t 3 y 3t 2 y + 6 t y 6 y = 0 ,
three linearly independent solutions
ELG 3120 Signals and Systems
Chapter 2
Chapter 2 Linear Time-Invariant Systems
2.0 Introduction
Many physical systems can be modeled as linear time-invariant (LTI) systems
Very general signals can be represented as linear combinations of delayed impulses.
ELG 3120 Signals and Systems
Chapter 1
Chapter 1 Signal and Systems
1.1 Continuous-time and discrete-time Signals
1.1.1 Examples and Mathematical representation
Signals are represented mathematically as functions of one or more independent variables. Here
ELG 3120 Signals and Systems
Chapter 4
Chapter 4 Continuous-Time Fourier Transform
4.0 Introduction
A periodic signal can be represented as linear combination of complex exponentials which
are harmonically related.
An aperiodic signal can be represented a
ELG 3120 Signals and Systems
Chapter 3
Chapter 3 Fourier Series Representation of Period Signals
3.0 Introduction
Signals can be represented using complex exponentials continuous-time and discrete-time
Fourier series and transform.
If the input to an LTI
ELG 3120 Signals and Systems
Chapter 5
Chapter 5 The Discrete-Time Fourier Transform
5.0 Introduction
There are many similarities and strong parallels in analyzing continuous-time and discretetime signals.
There are also important differences. For example
The Differential Operator
For a differentiable function, f (t), we define the differential operator, D, by:
Df =
We write D =
df
= f (t) = f .
dt
d
d
df
since D (f ) = (f ) =
.
dt
dt
dt
Then, D 2 f = D (D f ) = D (f ) = f , D3 f = f , and D n f = f (n) =
Differential Equations Use of Substitutions
A first-order differential equation of the form:
dy
= f (a x + b y + c) , where a, b and c are constants, b 0,
dx
can always be reduced to a differential equation with separable variables by means of the
substit
Differential Equations Reducible to First Order
Solve each of the following differential equations; if initial values are given, you only
need to find the particular solution that satisfies the differential equation and the initial values.
Pay attention t
Reduction of Order Solutions
Exercise 1: t 2 y 6 y = 96 t 6 , t > 0. Let y1 (t) = t 3 .
Then, t 2 y1 6 y1 = t 2 (6 t) 6 t 3 = 0 . Thus, y1 (t) = t 3 is a solution of the associated
y = v t 3 + v3t 2 , and
homogeneous equation. Let y (t) = v (t) t . Then
The Phase Line, Stability and Inflection Points
Consider the first-order autonomous ordinary differential equation:
dy
= f (y) .
dt
(1)
An autonomous differential equation is an ordinary differential equation which does not depend
on the independent varia
THE PROJECTILE PROBLEM
y = y(t)
Suppose a projectile is shot vertically upward from the surface of a
planet having radius R. The projectile has no engine; thus its motion
relies only on its initial velocity and gravity. We ignore air resistance.
Let y (t)
Rules for Undetermined Coefficients
We want to find a particular solution of a nonhomogeneous linear ordinary differential equation with
constant coefficients of the form:
ODE: L(y) = a n y(n) + a n 1 y(n 1) +
where
and
1.
2.
+ a1 y + a 0 y = g (x) ,
all
Variation of Parameters in a
Second Order Linear Ordinary Differential Equation
Consider the second order linear ordinary differential equation:
L (y) = a (t)
d2 y
dt
2
+ b (t)
dy
+ c(t) y = g (t) .
dt
(1)
We assume that a (t), b (t), c(t) and g (t) are c
Differential Equations Singular Solutions
Consider the first-order separable differential equation:
We solve this by calculating the integrals:
dy
= f (y)g (x) .
dx
dy
= g (x)dx + C .
f (y)
(1)
(2)
If y0 is a value for which f (y0 ) = 0 , then y = y0 wil
Reduction of Order
for a Second Order Linear Ordinary Differential Equation
Consider the second order linear ordinary differential equation:
L (y) = a (t)
d2 y
dt 2
+ b (t)
dy
+ c(t) y = g (t) .
dt
(1)
We assume that a (t), b (t), c(t) and g (t) are conti
Math 2800 - ODEs 1 Body Falling with Air Resistance
Consider an object that is falling under the force of gravity and we take into account the force of
air resistance (also called drag).
Let m be the mass of the object (kg), let t be time (sec), let g be