Prof. Readdy's Math 214 Differential Equations Course Homepage

### Announcements

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.

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

Take the Nerd Test.

### Course Diary

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 zn = 1. We saw the n powers of the primitive root of unity w = e2 π i/n gives the n solutions. (For example, the case n = 4 has w = e πi/2 = i, and the solutions to z4 = 1 are i, -1, -i and 1.) As motivation, Euler's formula e = 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).

##### Gian-Carlo Rota teaching differential equations at MIT. 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 d2y/dt2 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 = c1exp(at) cos (bt) + c2exp(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

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 = yh + yp, where yh is the solution to the homogeneous differential equation and yp 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(tn); 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 uc(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.

### Fun things to do when you are not thinking about Differential Equations

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

### Quotes

"In order to learn anything you need to see it three times". Professor Gian-Carlo Rota

"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.