Lecture Notes on
Differential Equations
Emre Sermutlu
ISBN:
Copyright Notice:
To my wife Nurten and my daughters Ilayda
and Alara
Contents
Preface
1 First Order ODE
1.1 Definitions . . . . . . . .
1.2 Mathematical Modeling
1.3 Separable Equations . .
1.4
Math 260
Assignment #1
Spring 2011
1. Solve the following matrix equation for a, b, c and d.
a b
c 3d d
=
c d
2a + d a + b
3 0
1 5 2
4
1
1
4
2
2. Consider the matrices A = 1 2 , B =
, C =
, D = 1 0 1
0
2
3 1 5
1 1
3 2 4
6 1 3
and E = 1 1 3 . Compute th
Math 260
Assignment #3
Spring 2011
In order to receive full marks, students must indicate the various steps taken.
1. Find the inverse of
1 2 3
(a) A = 2 5 3
1 0 8
1 0
2. Let A = 2 4
3 1
the following matrices (if possible)?
1 6
4
(b) A = 2 4 1
1 2
5
5
2
Math 260
Assignment #4
Spring 2011
webpage: http:/www.metu.edu.tr/bcanan
1. Evaluate the following determinants:
1
0
0
1
1 2
0
1 1 1
1
(a) 4 1 2
(b)
1
2
0 1 0
0
1
4 1 1
0
a b
2. If det
= 2, calculate:
c d
2
2 0
3c
a
+
c
1
(a) det c + 1 1 2a
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APPENDICES
A.1
Real Numbers and the Real Line
This section reviews real numbers, inequalities, intervals, and absolute values.
Real Numbers
Much of calculus is based on properties of the real number sy
Since the light is a wave motion, the two beams start out
in phase,
or in step. If the beams return in step, then the time
required for
the two trips is the same, but if they return out of step the
time
difference between the two trips is the time for hal
the ether has a length L o, and that a rod moving through
the
ether with its length aligned perpendicular to its direction
of
motion is unaltered, that is LJ. = L o Then Eq. 1-3.lc
gives the
time t:.t J. correctly, provided that we replace L by L J. or
by
shrinks, with its new length given by
(1-3.4)
we would find that
2Lo 1
Atn = t:.tJ. = ~ (1 _ V2/C2)1/2 (1-3.5)
This result would account for the failure to detect a time
difference
in the two paths in the Michelson-Morley experiment.
These explanations ha
away a negative experiment, they had no other detectable
consequence than to explain the effect for which they
were created.
But a train of thought was set into motion by the failure of
the
Michelson-Morley experiment to find an ether,
culminating in
the
same. This procedure takes account of the finite speed of
light,
and insures that there are no alterations in the clock's
behavior,
as there might be if they were carried to a central timing
station.
In anyone frame there is no doubt as to when two events
the rod will note that the light flashes in intervals of 2L/c,
assuming
the detector to be infinitely fast. An observer in the
laboratory notes that the path of the beam from the
moving
source to the moving mirror is not L long, but is from the
source
at
frame depends on where the events took place. From the
Lorentz
transformation, Eqs. 3-1.3, we note that
32 AN INTRODUCTION TO THE SPECIAL
THEORY OF RELATIVITY
t\ = 'Y(t1 - VxI/c2) and t'2 = 'Y(t2 - VX2/C2).
The difference between the time of the two event
Since there was no independent way of determining that
the
length of the two paths was identical, the instrument was
first
adjusted to yield a bright view to the observer and then
was rotated
by 90. By this means any difference between the parallel
and pe
can detect the absolute motion of an inertial reference
frame.
Thus there is no purpose in speaking of a velocity of light
except
with respect to the observer who measures it. We reason
further
that no one observer has a special place, superior to all
oth
it is true at speeds comparable to the speed of light. We
will
require of statements of relativity that they agree with
ordinary
experience in the limit of low velocities. Whenever an
experimental
test has been made of deductions from the special theory
o
Lorentz transformation yields the results of ordinary
experience
in the limit of low velocities.
By direct substitution of Eqs. 2-1.3 into the equation of
the
light sphere in the unprimed frame, Eq. 2-1.1, we find the
equation
of the light sphere in the p
time interval between flashes of 2t:.to which the rest
clock reads,
28 AN INTRODUCTION TO THE SPECIAL
THEORY OF RELATIVln
the moving clock is observed by the rest observer to take
a time
2y.:lto between flashes. The speed of a moving object is
always
less
an observer in the laboratory frame, for whom the rod
clock is
moving with speed v transverse to the orientation of the
rod,
measures the time interval between flashes to be 2M, and
that
these two are related by the equation
(1-5.1)
In the literature of r
Now consider the meaning of the rod clock experiment.
The
laboratory observer and the moving observer both have
rod
clocks. Each observer knows his clock to keep correct
time. Yet
whichever is considered to be the observer at rest notes
that the
moving cl
As a first step in the application of the general principle,
let
us suppose that the constancy of the speed of light is a
physical
law; that is, the speed of light is the same in every inertial
frame.
Then the speed of light is the same in every direction
1-5.1 Find 'Y for fJ = S/5, 4/5, 5/U, 12/IS. Use the right
triangle
relationships. If sin (J =fJ, then sec (J ='Y. [1.2, 1.67,
1.08, 2.6]
1-5.2 If fJ =I - x, then 'Y-2 =2x - x2 E! 2x for small x or
for fJ close
to I. Prove that the error in 'Y in making t
measurement of time. Moving clocks do not run at the
same rate
as clocks fixed in the laboratory. We shall see that moving
clocks
Tun slow.
Let us design a clock, taking account of our new view of
the
speed of light. If the speed of light is constant, and
to the source. Such a system is an oscillator, or a rod
clock. If the
light source is initially flashed, the light will travel down
the
stick and back with speed c and be detected; the detector
will
then trigger the light source to flash again. The time
i
on the ground who measures the velocity of an airplane
with respect to his frame will arrive at an entirely
different result
than the pilot reading his airspeed indicator and his
compass
heading. If these observers communicate by radio, their
results
are
length of each section is about 15 m yields an elapsed
time of
10-7 sec for the return trip. Thus a time measurement to
half
the period of oscillation of a light wave could measure the
transit
time to 10-15/10-7, or to I part in 108, as required. In the
a
define any two of the three quantities, the unit of length,
the
unit of time, or the speed of light, but not all three. At
present
it is simpler to define a fundamental unit of length and a
fundamental
unit of time, and to measure the speed of light, but
bright, while if out of step the two waves cancel, and the
view is
dark.
It is easy to see that this apparatus might have the
necessary
precision. Yellow light, of wavelength 600 millimicrons
(nanometers),
has a period of 2 X 10-15 sec, so that the time f