AP Statistics
Chapter 1-5 Review
1. Wetlands offer a diversity of benefits. They provide habitat for wildlife, spawning grounds for U.S. commercial fish, and
renewable timber resources. In the last 200 years, the United States has lost more than half its
Repeated Eigenvalues
1. Repeated Eignevalues
Again, we start with the real 2 2 system
.
x = A x.
(1)
We say an eigenvalue 1 of A is repeated if it is a multiple root of the char
acteristic equation of A; in our case, as this is a quadratic equation, the o
Modied Input
Way back when we introduced the language of system, input and re
sponse we decided that the right hand side of our equations wasnt always
the input. Sometimes it was a modied version of the input.
Example 1. Recall the heat diffusion equation
Unit Step Response: Post-initial Conditions
Quiz: Consider the equation
.
v + kv = u(t)
with rest initial conditions, v(0 ) = 0.
.
For the solution v(t) what is v(0+ )?
Choices:
.
a) v(0+ ) = 0
.
b) v(0+ ) = 1/k
.
c) v(0+ ) = 1
.
d) v(0+ ) = k
e) None
Initial Conditions
1. Introduction
Before we try to solve higher order equations with discontinuous or
impulsive input we need to think carefully about what happens to the so
lution at the point of discontinuity.
Recall that we have the left and right lim
Unit Step Response: Post-initial Conditions
Quiz: Consider the equation
.
v + kv = u(t)
with rest initial conditions, v(0 ) = 0.
.
For the solution v(t) what is v(0+ )?
Choices:
.
a) v(0+ ) = 0
.
b) v(0+ ) = 1/k
.
c) v(0+ ) = 1
.
d) v(0+ ) = k
e) None
Unit Step Response: Post-initial Conditions
Quiz: Consider the equation
.
v + kv = u(t)
with rest initial conditions, v(0 ) = 0.
.
For the solution v(t) what is v(0+ )?
Think about your answer and then look at the choices.
MIT OpenCourseWare
http:/ocw.mit
Generalized Derivatives.
Quiz: When you re a gun, you exert a very large force on the bullet over
a very short period of time. If we integrate F = m a = m x we see that a
large force over a short time creates a sudden change in the momentum,
m x . This is
Generalized Derivatives.
Quiz: When you re a gun, you exert a very large force on the bullet over
a very short period of time. If we integrate F = m a = m x we see that a
large force over a short time creates a sudden change in the momentum,
m x . This is
First order Unit Impulse Response
1. Unit Impulse Response
Consider the initial value problem
.
x + k x = ( t ),
x (0 ) = 0,
k, r constants.
This would model, for example, the amount of uranium in a nuclear reactor
where at time t = 0 we add 1 kilogram of
First order Unit Step Response
1. Unit Step Response
Consider the initial value problem
.
x + k x = r u ( t ),
x (0 ) = 0,
k, r constants.
This would model, for example, the amount of uranium in a nuclear reactor
where we add uranium at the constant rate
Poles, Amplitude Response, Connection to ERF
For our standard LTI system p( D ) x = f , the transfer function is W (s) =
1/ p(s). In this case the poles of W (s) are simply the zeros of the character
istic polynomial p(s) (also known as the characteristic
Existence and Uniqueness and Superposition in the General Case
We can extend the results above to the inhomogeneous case.
x = A(t)x (homogeneous)
(H)
x = A(t)x + F(t) (inhomogeneous),
(I)
where F (t) is the input to the system.
Linearity/superposition:
1.
Summary
In summary, the procedure of sketching trajectories of the 2 2 linear
homogeneous system x = Ax, where A is a constant matrix, is the fol
lowing.
Begin by nding the eigenvalues of A.
1. If they are real, distinct, and non-zero:
a) nd the correspon
Introduction
In this session we learn general results about the solutions of any n n
linear DE system (not necessarily constant coefcient).
First we will learn the some general theory for linear systems. This will
be familiar to you from our study of line
Sketching More General Linear Systems
In the preceding section we sketched trajectories for some particular lin
ear systems. They were chosen to illustrate the different possible geometric
pictures. Based on that experience, we can now describe how to ske
Introduction
In this session we introduce and develop the basic properties of au
tonomous 2 2 systems. In the next session we will see how to get key
information about the solutions to such a system directly from the DE it
self, without having to actually
The Normalized Fundamental Matrix
In the previous note we saw two main facts about the fundamental matrix:
x1
x1 x2
=
.
(1)
(t) =
x2
y1 y2
and
x = ( t ) ( t 0 ) 1 x0 .
(2)
Is there a best choice for fundamental matrix?
There are two common choices, each w
Sketching the Basic Linear Systems
In this note we will only consider linear systems of the form x = Ax.
Such a system always has a critical point at the origin.
We start by sketching a few of the simple examples, so as to get an idea
of the various possi
General Linear ODE Systems and Independent Solutions
We have studied the homogeneous system of ODEs with constant co
efcients,
x = A x ,
(1)
where A is an n n matrix of constants (n = 2, 3). We described how
to calculate the eigenvalues and corresponding
The Wronskian
We know that a standard way of testing whether a set of n n-vectors are
linearly independent is to see if the n n determinant having them as its
rows or columns is non-zero. This is also an important method when the nvectors are solutions to
The Existence and Uniqueness Theorem for Linear Systems
For simplicity, we stick with n = 2, but the results here are true for all
n. There are two questions about the following general linear system that
we need to consider.
x = a(t) x + b(t)y
x
a(t)
Unit Impulse Response: Post-initial Conditions
.
Quiz: Let w(t) be the solution to m x + k x = (t) with rest initial condi
.
tions. What is w(0+ )?
Choices:
.
a) w(0+ ) = 0
.
b) w(0+ ) = m
.
c) w(0+ ) = k
.
d) w(0+ ) = k/m
.
e) w(0+ ) = 1/m
f) None of the
Second order Unit Step Response
1. Unit Step Response
We will use the example of an undamped harmonic oscillator with in
put f (t) modeled by
.
m x + k x = f ( t ).
The unit step response is the solution to this equation with input u(t) and
rest initial c
Example: Simple Harmonic Oscillator
Example. Let f (t) = the odd square wave of period 2 with f (t) = 1 for
0 < t < 1. Use Fourier series to solve the DE
.
x + 9.1x = f (t).
(1)
Solution. From previous examples we know the Fourier series for f (t),
4
sin
Using Step Functions as Switches
Quiz: What is the equation for the function which agrees with f (t) be
tween a and b (assume a < b) and is zero outside this window?
Choices:
a) (u(t b) u(t a) f (t)
b) (u(t a) u(t b) f (t a)
c) (u(t a) u(t b) f (t)
d) u(t
General Case
It is actually just as easy to write out the formula for the Fourier se
ries expansion of the steady-periodic solution xsp (t) to the general secondorder LTI DE p( D ) x = f (t) with f (t) periodic as it was to work out the
previous example -
Integration with Delta Functions
Quiz: Compute
10
0
5(t + 1) + 3(t) + t2 (t 5) + t(t 20) dt.
Choices:
a) 0
b) 25
c) 28
d) 33
e) 48
f) 53
g) none of these
Pick what you think is the correct choice and then look at the answer.
MIT OpenCourseWare
http:/ocw.
Generalized Derivatives.
Quiz: When you re a gun, you exert a very large force on the bullet over
a very short period of time. If we integrate F = m a = m x we see that a
large force over a short time creates a sudden change in the momentum,
m x . This is
Integration with Delta Functions
Quiz: Compute
10
0
5(t + 1) + 3(t) + t2 (t 5) + t(t 20) dt.
Think about your answer and then look at the choices.
MIT OpenCourseWare
http:/ocw.mit.edu
18.03SC Differential Equations
Fall 2011
For information about citing