The links are to Paul's Notes, the section numbers refer to Boyce and DiPrima (10th edition).
WeBWorK set 1
Precalculus essentials
Differentiation essentials
WeBWorK set 2
Introduction to WeBWorK, Classifying DEs.
Definitions from Paul's notes (section 1.1, 1.2,
1.3 in Boyce and DiPrima).
direction fields (section 1.1)
Worksheet 1 on Classification of DEs
for the first day of class,
with solutions.
Here is an Introduction to WeBWorK.
Even if you have used WeBWorK in other classes this has some useful information.
NOTE: You login name and password are those of your LOUIE account (e.g. jws8).
Worksheet 2
for the second day of class.
Here are the scanned solutions.
Worksheet 3
for the third day of class. Here are the scanned solutions,
with the numbering shifted.
WeBWorK set 3
separable 1st order ODEs (section 2.2)
Interval of Validity,
or interval of existence, of solutions. (section 2.4)
Definitions of General Solution and Interval of Existence.
Worksheet 4
for the fourth day of class. Here are the scanned solutions.
Here are the scanned solutions of Friday's quiz.
WeBWorK set 4
Linear 1st order ODEs
(section 2.1)
Recipe to solve any first order linear ODE for y(x).
Here’s the recipe for y(t).
Worksheet 6
for the sixth day of class. Here are the scanned solutions. (Note that these are solutions to an old version of the worksheet.)
Worksheet 7
for the seventh day of class. Here are the scanned solutions.
WeBWorK set 5
1st order modeling (section 2.3).
Solving dy/dt = k(y-A) by inspection.
Group work for Friday, Sept. 15, worth 5 class points.
Here are the scanned solutions.
Worksheet 9
for the ninth day of class. Here are the scanned solutions.
Here is full page solution to problem 3, taken from solutions to an old exam.
WeBWorK set 6
Autonomous Equations and
Equilibrium Solutions
(section 2.5).
Euler's method (Section 2.7).
Worksheet 10
for the tenth day of class. Here are the scanned solutions.
Here is a derivation of the solution
to the logistic equation which is claimed in set 6, problem 5.
Most textbooks do separation of variables, with partial fractions to do the \(P\) integral. Gross!
Euler’s method
Euler’s method scene from Hidden Figures.
Here’s a pdf on how to do
Euler’s Method with a Spreadsheet.
I took this screen shot from the Hidden Figures
Euler’s method Scene, with a descrption of the Modified Euler’s method.
This method is also called
Heun’s method,
and it is the second method that our
Slope Field and Direction Field
applet uses.
I found this desciption of
the role math played in the Hidden Figures movie.
This is a more nerdy
blog about Katherine Johnson’s technical note which was mentioned in the film.
My video on how to do Euler's Method
with Google sheets.
WeBWorK set 7
Exact ODEs (Section 2.6).
These web sites might be helpful:
CalcPlot3D, and
Desmos Graphing Calculator.
Monday, Sept. 25: Worksheet 12
for the twelth day of class. Here are the scanned solutions.
Wednesday: Worksheet 13
for the thirteenth day of class. Here are the scanned solutions.
Friday, September 29, will be a review day, and Monday,
October 2 will be Exam 1 covering WebWorK sets 1-7.
The worksheet for Friday, Sept. 29 is printed out. Here are the scanned
solutions.
WeBWorK set 8 This is no longer assigned.
WeBWorK set 9
Basic Concepts
and Real, Distinct Roots of the characteristic equation (section 3.1)
Worksheet 14
for Wednesday, 10-4. Here are the scanned solutions.
Here is a Desmos Graph of the general solution to \(y'' + y'-2y = 0\).
Find the solution to the IVP \(y'' + y'-2y = 0, \quad y(0) = 2, y'(0) = -1\) and check with
this Desmos graph
Worksheet 15
for Friday, October 6. Here are the scanned solutions.
Note: We only did problems 1 and 2 on Friday. The other problems
are covered on Monday's group work.
WeBWorK set 10
Fundamental Solution Sets,
Linear independence, the Wronskian, and
Abel's Theorem (section 3.2).
Worksheet 16
for Monday, October 9. Here are the scanned solutions.
Summary of the Theory of Linear Homogeneous ODEs.
In lieu of a worksheet, we will have everybody work on their version of problem 4.
For problem 5, use the Theory of Linear Homogeneous ODEs pdf, and these two facts about determinants:
(1): \(\text{det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \text{det} \begin{bmatrix} a & c \\ b & d \end{bmatrix} = ad-bc\).
(2): Two vectors \( \langle a, b\rangle\) and \( \langle c, d\rangle\) are linearly independent if and only if \(\text{det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \neq 0\).
For problem 6, use Abel's theorem, which says that if \(y_1(t)\)
and \(y_2(t)\) are any two solutions to
\( y''+ p(t) y' + q(t) y = 0,\)
then \( W_{y_1, y_2}(t) = c \ \text{exp}\left (-\int p(t) dt \right) \)
for some constant \(c\).
The first order ODE version of Abel's theorem is that any solution
to \(y' + p(t) y = 0\) satisfies \(y(t) = c \ \text{exp}\left (-\int p(t) dt \right) \)
for some constant \(c\).
WeBWorK set 11
Complex roots of the characteristic equation, and
Repeated roots of the characteristic equation. (section 3.3, 3.4, 4.1)
Euler equations (section 5.4).
Worksheet 17
for Friday, Feb. 25 and Monday, Feb. 28. Here are the scanned solutions.
I make several videos to help with this problem set,
and to justify the rules for solving Linear Homogeneous ODEs with Constant Coefficients (LHODECCs).
YouTube: Set 11, problem 1
(Solving \(y'' = k^2 y\) by inspection.)
YouTube: Set 11, problem 5
(A messy IVP)
YouTube: Multiple roots of the characteristic equation
(Jusification of the rule)
YouTube: Complex conjugate roots of the characteristic equation
(Jusification of the rule)
YouTube: Euler's Formula
(Proof of \(e^{it} = \cos(t) + i \sin(t)\) using ODEs.)
YouTube: Hyperbolic Sine and Cosine functions
that can be used to solve \(y'' = (\text{positive constant}) \cdot y\).
Desmos graphs of
solutions to
\(y'' = \text{constant} \cdot y\).
Taylor series of exp, cos, sin, etc.
Here is a pdf about how to solve \(y'' = \text{constant} \cdot y \ \) by inspection.
WeBWorK set 12
Linear nonhomogeneous ODEs (section 3.5, 4.3)
The Method of Undetermined Coefficients (included in sections 3.5, 4.3)
Worksheet 18
for Wednesday, October 18.
Here are the scanned solutions.
Here is a pdf about the method of undetermined coefficients.
I made videos during the Spring 2020 lockdown for the rest of the class. The first set is here:
Set 12, Why is y = yh + yp?
Set 12, problem 5 (during COVID Spring) = problem 6 (this semester). Here is a link to the desmos graph for the
solution to this problem.
Set 12, problem 8
Set 12, problem 10.
Set 12, problem 11
Here are the scanned solutions to Friday's group work on Undetermined Coefficients.
Here is a
table from Boyce and DiPrima (10E) about how to choose
the form of \(y_p = Y\).
WeBWorK set 13
Worksheet 24, Exam Review,
for Friday, October 27. Here are the scanned solutions.
Exam 2 will be on sets 9-13, on Monday, October 30.
WeBWorK set 14
Worksheet 25.5, Taylor Series Review,
for Friday, November 3.
Here are the scanned solutions.
WeBWorK set 15
WeBWorK set 16
WeBWorK set 17
Wednesday, November 22:
The solution to \(\bf x' = A \bf x\) ,
Review for Final Exam
Videos
The Final Exam is scheduled for
Second Order Modeling, especially modeling of
Mechanical vibrations (sections 3.7, 3.8)
Worksheet 21, Undamped Oscillators,
for Monday, October 23. Here are the scanned solutions.
Videos
Set 13 introduction. Mass on a spring
Set 13, computing omega_0
Set 13, problem 1
Set 13, problem 2 introduction
Set 13, problem 2 finished
Set 13, problem 3
Set 13, problem 3 desmos
.
Here is the link to the desmos graph:
polar coordinates for oscillators.
Set 13, damped oscillators
Set 13, damped oscillator graphs with Desmos graphing calculator
.
Here are links to the three desmos graphs discussed in the video:
underdamped oscillator,
overdamped oscillator, and
critically damped oscillator.
Set 13, problem 6
Review: Second Order Modeling
NIST web page about the Quality Factor,Q
Worksheet 23, Driven Oscillators,
for Wednesday, October 25. Here are the scanned solutions.
Desmos animation of a
Shaken Mass on a Spring.
Desmos graph of the
Resonance Curve,
showing Full Width at Half Maximum Power.
Review of Power Series
and Taylor series (section 5.1).
Worksheet 25, Power Series Review,
for Wednesday, November 1.
Here are the scanned solutions.
Here's an animated gif
of the Taylor Polynomials that approximate \(\displaystyle f(x) = \frac{1}{1+x^2}\).
Note that \(T_5 = T_4\),
and \(T_5\) is actually a polynomial of degree 4, not 5.
Videos
Set 14, problems 1-5
Set 14, problems 7 9 10
Set 14, problem 6
Set 14, problem 8
Series Solution to 2nd order ODEs (sections 5.2 and 5.3).
Worksheet 26, Series Solution to \(y' +2y = 0\),
for Friday, April 1.
Here are the scanned solutions.
Videos
Set 15, problem 2
Set 15, problem 1
Set 15, solving recurrence relations This is useful for problems 2, 3, 4, and 5.
Near the end the video shows how to do problem 4 (finding polynomial solutions with series methods)
and gives a hint for problem 5
(getting an expression for \(c_{2n}\) or \(c_{2n+1}\) as a function of \(n\).)
Set 15, problem 6 sinc function
Set 15, problem 6
Set 15, more like problem 1
Systems of Linear Equations(not ODEs),
Matrices and Vectors,
Eigenvalues and Eigenvectors and
Introduction to Systems of Linear ODEs (sections 7.1, 7.2 and 7.3).
Worksheet 27, Systems of First Order ODEs,
for Monday, November 13.
Here are the scanned solutions.
Worksheet 28, Eigenvalues and Eigenvectors,
for Wednesday, November 15 and Friday, November 17.
Here are the scanned solutions.
Videos
Set 16, Introduction to Systems
of 1st Order ODEs, problems 1-4
Set 16, More about AB ≠ BA
Set 16, problems 5-7
Set 16, problems 8, 14 and 15
(Eigenvalues and Eigenvectors, part 1).
Set 16, Eigenvalues and Eigenvectors, Part 2
How to compute eigenvalues and eigenvectors.
Set 16, Eigenvalues example 1
(Real eigenvalues.)
Set 16, Eigenvalues example 2
(Complex eigenvalues.)
Set 16, Eigenvalues example 3
(Eigenvalues of a matrix with parameters.)
Here are figures of some vector fields and their eigenvalues and eigenvectors:
eigenvalues 1 and 2,
eigenvalues 1 and -2,
irrational eigenvalues,
pure imaginary eigenvalues,
complex eigenvalues,
repeated eigenvalues 0, 0,
repeated eigenvalues -1.5, -1.5, and
A = 2I with repeated eigenvalues 2, 2.
Here is the Mathematica notebook, eigenvectors.nb,
that made these figures.
Solutions to systems of Linear ODEs and the
Phase Plane.
Solutions to \({\bf x}' = A{\bf x}\), where \(A\) has
Real Eigenvalues,
Complex Eigenvalues
and Repeated Eigenvalues
(sections 7.5, 7.6, 7.8).
Monday, November 20: Constructing a 2x2 matrix with a given pair of eigenvalues is easy with the
eigenvalue hack described in this
worksheet 29.
Here are the scanned solutions.
Practice with this Worksheet 29.5. The solutions are the second page of the
these scanned solutions.)
Here is a Mathematica Notebook that checks the solution to problem 3:
ws29p3solution.nb
Monday, November 27: Repeted eigenvalues, and normal modes (like problem 10):
Practice this with Worksheet 30. The solutions are the second page of the
these scanned solutions.)
Videos
Set 17 intro to problems 1 to 9
Set 17 example 1
(Matrix has real, distinct eigenvalues.)
Set 17 example 1 computer pics
Set 17 problem 3, slow method
(Complex eigenvalues, using the book's method)
Set 17 problem 3, swift method
(Complex Eigenvalues, using the Swift method.)
Set 17 example 2 (problem 3) computer pics
(WeBWorK problem 3 computer pics)
Set 17 example 3
(Matrix has repeated eigenvalues. Like WeBWorK problem 6.)
Set 17 examples 4 and 5
(Matrix has repeated, nonzero eigenvalues. Like WeBWorK problem 7.)
Set 17 examples 4 and 5 computer pics
(Computer pictures for two matrices with repeated, nonzero eigenvalues.)
Set 17 problem 10
(Normal modes in a mass-spring system.)
Set 17 problem 10 computer animation
Worksheet 32, Final Review; 1st Order ODEs
for Monday, December 4.
These are problems 4 and 7 from the Sample Final Exam (on BbLearn).
The solutions are posted to BbLearn.
Worksheet 33, Final Review; Higher Order ODEs
for Wednesday, December 6.
Here are the scanned solutions.
The review videos for the whole course use the ODE \(y' = -3(y-20)\) as an example for everything!
Review: Intro to DEs and Separable 1st order ODEs
(WeBWorK sets 2 and 3.)
Review: Linear 1st order ODEs
(WeBWorK set 4.)
Review: Modeling and Autonomous ODEs
(WeBWorK sets 5 and 6.)
Review: Euler's Method
(WeBWorK set 6.)
Review: Exact ODEs
(WeBWorK set 7.)
Review: Linear ODEs with constant coefficients, and Undetermined Coefficients
(WeBWorK sets 9-13.)
Review: Series Solutions
(WeBWorK sets 14 and 15.)
Review: Systems of Linear 1st Order ODEs
(WeBWorK sets 16-18.)
Monday, December 11, 12:30-2:30
Instructor Information
Jim Swift's home page
Department of Mathematics and Statistics
Bb Learn
NAU Louie
NAU Home Page
e-mail: Jim.Swift@nau.edu
