Prof. Readdy

University of Kentucky

Spring 2017

Lecturer: Prof. Readdy, 825 POT, 859-257-4680, margaret.readdy@uky.edu

Office Hours: By appointment

Math Dept. Secretary: phone 257-3336

Please refer to syllabus for further details.

Course Schedule (approximate)

Review 0. Prerequisites for the course in a nutshell.

Exam 1

High 100

Low 42

Average 76

90-100 A; 80-89 B; 70-79 C; 60-69 D; Below 60 E

Friday, March 10th is the last day for you to ask questions regarding Exam 1 grading. If you have a question, write it on your exam and hand it in to me in class on Friday.

I will be in my office on Friday, March 10th from 11-11:30 am and then 1:00 - 1:30 pm.

I will be using your Exam 1 grade when I post your midterm grade.

Special Mathskeller hours for Math 214 in the Klein room! (See below under "Help resources")

Take the Nerd Test.

**Week 1 **

**Wednesday, January 11th **

Tacoma Narrows Bridge

Millenium Bridge on opening day

The Volgograd Bridge (Russian: Волгоградский мост) in 2010. This is located over the Volga River in Volgograd, Russia. The Germans and Swiss were called in to save the day. They developed and installed semi-active tuned mass dampers on the bridge.

Demonstration of aeroelastic flutter

Today we spoke about the Tacoma Narrows Bridge, the (former wobbly) Millenium Bridge and the Volgograd Bridge. We reviewed polynomial approximations of functions from Calculus, Taylor series, important examples (exp(x), sin(x), cox(x), 1/(1-x)) and termwise differentiation.

We then spoke about the the natural numbers ℕ = {1, 2, 3, ...}, the integers ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}, the rational numbers ℚ = {a/b where a and b are integers, b neq 0}, the real numbers ℝ and the complex numbers ℂ= {a + bi : a, b ∈ ℝ}. Next time: More on complex numbers.

Homework: Read section 5.1 of Boyce and DiPrima and do (1, 6, 9, 11, 14, 17). Due Wed Jan 18 Read Paul's Online Math Notes, Complex Number Primer. If you prefer, you can read about complex numbers in Seeley, Calculus of One Variable, chapter 1.

**Friday, January 13th **

Today we spoke about operations on complex numbers (addition, multiplication, multiplicative inverse, conjugate), modulus (distance to the origin) and argument (angle). We discussed the fundamental theorem of algebra and proved Euler's formula. We showed the polar version of complex numbers reveals the geometry behind multiplying two complex numbers (the moduli = distance from the origin multiply and the arguments = angles add). Next lecture we will continue to emphasize the algebraic and geometric viewpoints of the complex numbers and then start chapter 1 in Boyce--DiPrima.

Homework: Read Paul's Online Math Notes, Complex Number Primer. If you prefer, you can read about complex numbers in Seeley, Calculus of One Variable, chapter 1. From Seeley do Page 381 (1bef, 2), Page 386 (Exercise 70), Page 388 (2abc, 4a, 7). Due Fri Jan 20 Also, read chapter 1 of Boyce & DiPrima for Wednesday's lecture next week.

**Week 2 **

**Monday, January 16th **

No class.
Martin Luther King Day.

**Wednesday, January 18th **

Today we finished talking about complex numbers
by looking at the n solutions to
z^{n} = 1.
We saw the n powers of the primitive root of unity
w = e^{2 π i/n} gives the n solutions.
(For example, the case n = 4 has w = e^{ πi/2} = i,
and the solutions to z^{4} = 1 are i, -1, -i and 1.)
As motivation,
Euler's formula e^{iθ} = cos θ + i sin θ
will allow us to describe the movement of interesting
oscillating phenomena, such as the movement of slinkies.

We then switched gears and spoke about differential equaitons in general. This included an example of a slope field and discussing what happens after changing the initial condition. We then discussed section 2.1 solving linear equations using integration factors. Right now this is a trick, but next time we will see it is a technique.

Homework: Read chapter 1 of Boyce and DiPrima (if you haven't already) and do §1.1 (15, 18, 19) and §1.3 (7, 9, 11). Due Wed Jan 25

**Friday, January 20th **

Finished 2.1 solving linear equations using integration
factors.
We then started separable equations.

Homework: Read §2.1 and do (13--18, 19*, 20). Due Fri Jan 27

**Week 3 **

**Monday, January 23rd **

Today we finished separable equations §2.2.
Began §2.4 Differences between linear and nonlinear
equations.

Homework: Read §2.2 and do (2, 6, 9, 10, 14, 20, 32ab, 35ab). Due Mon Jan 30

Quiz 2 was returned today. If you scored 5 or below out of 10 on Quiz 1 or Quiz 2, please view the video below.

I also mentioned how MIT Professor Gian-Carlo Rota said that in order to learn anything you need to see it three times. So for example, in this class first you read the text (1), then you go to lecture (2) and then you do your homework (3).

Photo by Zareena Hussain.

**Wednesday, January 25th **

Finished §2.4 today.
I also gave an example where an implicit solution
can be turned into an explicit one by completing the square.
We started §2.5 Autonomous Differential Equations today.
We covered exponential growth. To be continued...

Homework: Read §2.4 and do (1-11 odd, 13, 16, 22, 25). Due Wed Feb 1

**Friday, January 27th **

We covered exponential growth, equilibrium solutions and
logistic growth today. We saw how one can analyze the behavior
of the solutions without even solving the differential equation.
This includes being able to sketch typical solution curves.
The theory from §2.4 comes in play.
Today we saw that the phase line was a convenient
way to understand the general solutions of
an autonomous system without having to solve
it explicitly.
Many of the ideas from Calculus (critical points
via setting dy/dt =0, looking at where dy/dt is
increasing and decreasing, and computing
d^{2}y/dt^{2} to find the
point(s) of inflection) helped to sketch
representative solution curves of the general
solution.

To solve these types of differential equations, partial fractions come to the rescue.

Homework: Read §2.5 and do (3, 4, 5, 9, 22). Due Fri Feb 3

**Week 4 **

**Monday, January 30th **

We finished §2.5 today, ending with a
commercial for autonomous systems.
We then began speaking about §2.6, exact
differential equations.
We will continue the discussion on Wednesday.

Homework: Read §2.6 and do (1, 4, 5, 7, 11, 15, 18, 25, 26, 27, 32). Due Wed Feb 8

**Wednesday, February 1st **

Continued 2.6 today, showing how to convert an (almost!)
exact equation to an exact one using an integration factor.
We will finish up an example next time,
and then it is on to
2.7.

**Friday, February 3rd **

Finally finished exact differential equations today.
We then spent some time using
GeoGebra's slope field plotter.
The next section,
§2.7 Numerical approximations: Euler's method
outlines exactly how these on-line plotters work,
by following the slope field step by step.

Homework: Read §2.7 and do (1, 4, 6, 7, 11a). Due Fri Feb 10

**Week 5 **

**Monday, February 6th **

Finished lecturing on Numerical approximations: Euler's method today.
I then spoke about §2.9 First order difference
equations.
This included a discussion of the logistic difference equation
and bifurcation.

Homework: Read §2.9 and do (2, 3, 4, 6, 8). Due Mon Feb 13

**Wednesday, February 8th **

Today we finished §2.9 showing how to
iterate the logistic difference equation geometrically.
We saw that as rho gets larger, the behavior of the system
goes from converging, to bifurcations to chaos.
Wikipedia has a nice movie showing the
cobweb diagram of logistic map
. (wikipedia)

We then started to discuss chapter 3 Second order linear equations. We saw that for homogeneous equations with constant coefficients one can guess the solution y = exp(rt). This gives the characteristic equation of the differential equation. We covered the case of distince roots and started complex roots.

Homework: Read §3.1 and do (2, 3, 4, 9, 10, 14, 17, 19, 22, 24). Due Wed Feb 15

**Friday, February 10th **

We began by finishing an example of a degree
2 constant coefficient homogeneous diff'l eqn
having a characteristic polynomial with complex
roots (section 3.3).
We saw how this gives solutions of the form
y = c_{1}exp(at) cos (bt) + c_{2}exp(at) sin (bt).
I also gave a preview of what to do when we have repeated roots.
We will return to this when we cover section 3.4.
At the end I hinted how the Wronskian will help us determine
whether or not solutions to a differential equation are
linearly independent or dependent.

I distributed a paper copy of the Exam 1 review. sheet.

Quiz 5 today on Week 4 material!

Homework: Read §3.3 and do (3, 4, 9, 15, 17, 18). Due Fri Feb 17

**Week 6 **

**Monday, February 13th **

Today we spoke about the existence & uniqueness theorem in action,
linear operator, principle of superposition,
linear independence and dependence, fundamental set of solutions,
connections with Wronskian.
We still have some of section 3.2 to cover for next time.

Homework: Read §3.2 and do (1, 2, 6-9, 11-14, 16, 22, 24, 29, 33). Due Mon Feb 20th

**Wednesday, February 15th **

Wrapped up Wronskian today. Started section 3.4 today.

**Friday, February 17th **

Review for Exam today.

**Week 7 **

**Monday, February 20th **

Exam I today.

Material: Up through section 3.3

Please bring photo ID.

**Wednesday, February 22nd **

Today we finished
3.4 Repeated roots and reduction of order.
We saw that in order to solve these sorts
of differential equations, you need to be given
one of the solutions in order to find the second.
We also started discussing
section 3.5 nonhomogeneous equations.

Homework: Read §3.4 and do (4, 13, 23, 25, 27, 28). Due Wed March 1st

**Friday, February 24th **

We continued with §3.5 today
nonhomogeneous equations and
the method of undetermined coefficients
(also known as the fine art of guessing).
We saw that a general solution is
of the form
y = y_{h} + y_{p},
where
y_{h}
is the solution to the homogeneous differential
equation and
y_{p}
is the solution to the original differential
equation.
The geometry behind this is that the
homogeneous equation determines a subspace.
Making the equation nonhomogeneous
determines an affine subspace,
which is just
a translation of the original
subspace.

We did many, many examples today, including when the right-hand side of the differential equation is an exponential, polynomial and a trig function, and what to do when you accidentally pick up a homogenous solution. To be continued...

Homework: Read §3.5 and do (1, 3, 4, 5, 9, 10, 20, 29). Due Fri March 3rd

**Week 8 **

**Monday, February 27th **

We finished up 3.5 today by solving a baby
version of a differential equation for
the bridge vibration.

**Wednesday, March 1st **

§3.6 Variation of parameters today.
This is a nice technique to use when the
method of undetermined coefficients (aka
guessing) fails.

Homework: Read §3.6 and do (1, 3, 5, 7, 14, 18, 29) Due Wed March 8th

**Friday, March 3rd **

Quiz today on Week 7 material.
Did another example of variation of parameters today.
We then began
§3.7 Springs (Mechanical and electrical vibrations).
We began looking at the differential
equation of an ideal spring,
first without friction, and then
with friction.
This was a good review of
degree 2 constant coefficient
homogeneous equations.
Continuing on Monday,
we will see how
we get different behavior
(no friction/infinite oscillation, underdamped/ocillations of decreasing
amplitude, overdamped or
critically damped)
depending upon the roots of the characteristic equation.

Homework: Read §3.7 Springs (Mechanical and electrical vibrations) and do (2, 3, 5, 6, 7, 9). Due Fri March 10th

**Week 9 **

**Monday, March 6th **

Finished springs today: Friction (c>0) with
subcase B: two real roots and subcase C: double root.
I then asked the practical question of
which case would be the best when designing
shock absorbers for a car? An amusement park ride?
We ended by rewriting the complex root case using a single
trig function.

We then moved on to §3.8 forced vibrations.

Two tuning forks making amplitude modulation, also know as beat

Homework: Read §3.8 Forced vibrations and do (1--4, 5, 7, 9) Due Mon March 20th

**Wednesday, March 8th**

Started discussing chapter 4 on higher order linear eqns.
Exam 1 was handed back.

Homework: Read § 4.1 Higher order linear eqns and do (2, 6, 7, 13, 17, 21). Due Wed March 22nd

** Friday, March 10th**

Quiz today.
Here is the
answer key.
Continued higher order linear equations today,
including how to find rational roots.
Enjoy your break!

Homework: Read § 4.2 (Higher order) Homogeneous eqns with constant coeffs and do (1, 2, 10, 11, 17, 19). Due Fri March 24th

**Spring Break Week **

**Week 10 **

**Monday, March 20th**

Finished higher order linear equations today
(4.1, 4.2 and 4.3) and began
5.2 series solutions near an ordinary point, part I.

Homework:
Read § 4.3 (Higher order) Method of undetermined coeffs and do (3, 5, 8).

Due Mon March 27th

**Wednesday, March 22nd**

Finished
§5.2 Series solution near an ordinary point, part I.

Homework: Read § 5.2 Series solution near an ordinary point, part I and do (2, 3, 5, 8, 13, 15a, 17a, 21, 22) Due March 29th

** Friday, March 24th**

§5.3 Series solution near an ordinary point,
part II today.

Homework: Read § 5.3 Series solution near an ordinary point, part II and do (1, 3, 7, 8, 12, 17) Due Wed March 29th

**Week 11 **

**Monday, March 27th**

Today we began chapter 6, the Laplace Transform and
started to compute the Laplace transform of some
fundamental functions.
We also verified the Laplace
transform is a linear operator.
We then showed how to use the Laplace transform
to reduce an initial value problem into an algebraic
expression. Amazing! And then solve the initial value
problem in a painless way. Great!

Introduction to the Laplace transform Part 1: Definition; Laplace transform of fundamental functions.

Intro to Laplace transform Part 2: Linearity.

Laplace transform and IVP Part 1: L(f') and solving a first order IVP.

Homework: Read § 6.1 The Laplace Transform and do (1, 2, 5, 7, 15, 18, 21, 23, 26). Due Wed April 5th

**Wednesday, March 29th**

Reviewed for Exam II.

Here is the
Quiz 8 answer key.

** Friday, March 31st**

Exam II Today.

Material: section 3.4 through section 5.3.

**Week 12 **

**Monday, April 3rd **

No class.

**Wednesday, April 5th**

Continued with solving initial value problems today using the Laplace transform.

Homework: Read § 6.2 Solutions of IVP and do (1, 2, 3, 5, 7, 8, 12, 13, 17, 19, 21). Due Wed April 12th

Laplace transform and IVP Part 2: Philosophy behind solving an initial value problem using the Laplace transform; a first example.

Laplace transform and IVP Part 3:
"I love Euler"; Laplace transform of f ' ' (second derivative of f); Example of 2nd order IVP.

Laplace transform and IVP Part 4: L(t^{n}); Start of computation of
L(exp(at)*f(t)).

Laplace transform and IVP Part 5: Example of second order IVP with shift.

** Friday, April 7th **

Today we introduced the Heaviside function u_{c}(t) today.
This is a type of step function.
We also began using the
Heaviside function to solve differential equations.
How handy! How easy!

Read § 6.3 Step functions (aka the Heaviside Function) and do (1-9 odd, 12, 3, 15, 18). Due Fri April 14th

**Week 13 **

**Monday, April 10th **

We continued to use the Heaviside function to
solve differential equations today.
We then began speaking about the Dirac function,
named after the physicist Paul Dirac.
To be continued...

Homework: Read §6.4 and do (1, 4, 6, 7, 9). Due Mon April 17th

**Wednesday, April 12th**

We continued with §6.5 Dirac function today.
We then began §6.6 The Convolution Integral.
We finished by looking at some Doc Edgerton photos.
See
Pictures
by
Harold "Doc" Edgerton.

Homework: Read §6.5 The Dirac function and do (1, 3, 4, 6, 11). Due Wed April 19th

** Friday, April 14th **

Continued with the convolution integral today.
We saw how the Laplace transform takes convolution
to product, that is,
L(f*g) = F(s) G(s). Next time we will discuss
what to do with the annoying boss who likes to change
the forcing function.

Quiz today!

Homework: Read § 6.6 The Convolution Integral and do (2, 4, 9, 11, 15, 16, 19, 20). Due Fri April 21st

**Week 14 **

**Monday, April 17th **

Finished convolution integral today. We then began
looking at homogeneous linear systems. We first
solved an example using the Laplace transform -- not
hard, but very long. We then used the answer,
a vector combination of exponentials, to motivate a
linear algebra approach.
To be continued...

Read § 7.5 Homogeneous linear systems with constant coefficients and do (2, 3, 6, 15, 16). Skip direction fields for exercises 2, 3 & 6!

**Wednesday, April 19th**

We derived the idea of the eigenvalue and eigenvector from linear algebra
today. We saw that to solve a system of the form
**x'(t)** = A **x(t)**, it was enough to find the eigenvalues and
eigenvectors of the matrix A. We did this by looking at
the roots of det(A - λI) = 0, that is, the characteristic polynomial.
To find an associated eigenvector **v**, we needed to have
A **v** = λ **v**, that is,
(A - λ I)**v** = 0. We do this by subtracting
λ off of the diagonals of A, and then determining what
combination of the columns of this matrix is the zero vector.

We also practiced using the phase diagram software. We realized it is an art and can be misleading. Beware!

All of this (eigenvalues, eigenvectors, eigenspace) can be generalized to systems with more than 2 equations. However, we will stick with systems with 2 equations.

** Friday, April 21st**

**Week 15 **

**Monday, April 24th **

Complex eigenvalues and eigenvectors today.

**Wednesday, April 26th**

Here is the Review 3 and More review

** Friday, April 28th **

**Week 16 **

**Monday, May 1st **

Final Exam 1:00 pm - 3:00 pm

CB 214 (the usual place)

The final exam is cumulative.

Math 214 Mathskeller Hours:

Open Monday through Friday 9 am to 5 pm

Special Math 214 Mathskeller hours in the Klein Room:

Spring 2017 hours.

Saturday March 22nd, 1-5 pm, downtown Lexington.

"If you have time to look at facebook then you have time to read
the textbook." Professor Margaret Readdy

Last updated: Thursday, April 20, 2017.