Math 307 - Introduction to Differential Equations - Video Lecture Materials

Heather Lee (heathml@uw.edu)



Lectures

The textbook referenced is Boyce and DiPrima's Elementary Differential Equations.

It is essential that you watch the videos, not just read the notes. Minor mistakes/typos in the videos are corrected in the notes, so make sure you check the notes that are posted here.

The videos on Youtube and Panopto are identical, and they are captioned (not by me). You can also watch all videos from the Youtube playlist or the Panopto playlist. Panopto videos are restricted to those with an UW NetID. Edit: starting Aug. 20, 2021, the videos will no longer be publicly available.

No. Video Topics and Comments Notes              
L1 Youtube
Panopto 		Textbook reading: 1.1
Keywords: modeling, slope field, long term behavior
Modeling examples: population growth and decay
Objectives:
    (1) Translate from words to equations, e.g. translating information about the amount of change in a quantity per unit time as a derivative.
    (2) Draw a slope field from a differential equation.
    (3) Understand the geometric meaning of a solution, i.e. solutions are integral curves to the slope field.
    (4) Qualitatively describe the solution by looking at the slope field, e.g. predict the future by describing the long term behavior of the solution curve.
 Notes
for L1-L3
L2 Youtube
Panopto 		Textbook reading: 2.7, 1.2
Keywords: Euler's method, separable equations, doubling time
Modeling examples: population growth and decay
Objectives:
    (1) Understand the idea of Euler's method (iterative tangent line approximations). Given a 1st order differential equation and an initial condition, be able to use Euler's method to compute the value of the solution at a future time.
    (2) Solve simple separable equations like the example provided in this lecture. We'll learn more about separable equations in future lectures, so no worries if you don't feel like you know them really well in this intro lecture.
    (3) Understand the relationship between the doubling time and the exponential growth rate.
Extra links: Euler's method with Python, eulerExp.py and explanations, written by William Casper. Also see the numerical resources section.
L3 Youtube
Panopto 		Note: turns out WA DoH went back and corrected some of their numbers later on (now 3/20=1751, 3/21=1979, 3/25=3227). One can now see all previous dates' data (when I was making the videos, I could only see the data from the day of, so I recorded the numbers on the above 3 dates at the end of each day). The corrected numbers make our model look a bit better, but it still has all those issues I mentioned especially for long term estimates. Note another way to estimate r when you have several days' numbers is to look at the doubling time.
Textbook reading: 2.7, 1.1, 1.2
Keywords: Euler's method cont'd, separable equations
Modeling examples: population growth and decay, population growth with harvesting
Objectives:
    (1)The discussion about the errors of Euler's methods and other approximation methods are completely optional.
    (2) All mathematical models are approximations to the real scenario and can have minor or major drawbacks, know what they are.
    (3) The whole process: modeling by translating words into equations, draw slope fields and solution curves to qualitatively describe the behavior of the solutions, solve the equation numerically or analytically to obtain more quantative information.
L4 Youtube
Panopto 		Textbook reading: 1.1, 1.2
Keywords: equation of motion, laws for forces
Modeling examples: vertical motion experiencing gravity and drag
Objectives:
    (1) Equation of motion: m dv/dt = force
    (2) The magnitude of the force due to gravity near the surface of the earth = mg.
    (3) The magnitude of the drag force depends on various factors (the two common cases are proportional to v and v^2, respectively). Students are not responsible to figuring out which case it is; the problems given to you by the instructors will indicate that information.
    (4) Students should be able to figure out the sign of the force based on chosen sign convention and the direction of the force.
Extra links: drag of a sphere or baseball baseball (briefly discussed in this lecture), and in case you want a job at the UW wind tunnel 		Notes
for L4, L5(part a)


Notes
for
L5(part b)
L5 Youtube
Panopto 		Note:there is a typo in example 2, y'=x+y, which is NOT separable (I said it correctly but wrote down the wrong thing). This is corrected in the notes.
Textbook reading: 1.1, 1.2, 2.2
Keywords: equation of motion cont'd, separable equations
Objectives:
    (1) Describe the solutions to the equations of motion.
    (2) Identify separable equations, and find implicit, and sometimes explicit, solutions to them.
    (3) You don't need to be able to draw the slope field and solution curves exactly by hand for separable equations, but you need to be able to identify some qualitative features about them, e.g. where the slope is vertical, where the initial point is, pick out the curve representing the solution to an initial value problem, etc.
L6 Youtube
Panopto 		Textbook reading: 2.1
Keywords: linear first order equations
Objectives:
    (1) Identify linear equations
    (2) Introducing the method of integrating factors.
    (3) You don't need to be able to draw the slope field and solution curves exactly by hand for linear equations, but you need to be able to identify some qualitative features about them, e.g. where the slope is horizontal, where the slope is positive or negative, etc.
Notes
L7 Youtube
Panopto 		Textbook reading: 2.1 (optional: example 3 of 2.4)
Keywords: linear first order equations cont'd
Objectives:
    (1) Solve any linear 1st order equations using the method of integrating factors. Note that students won't be asked to carry out derivations of general formulas (i.e. entirely symbolic formulas without numbers) as done at the beginning of this lecture, but you should watch it once to get the idea. To solve equations, you can either carry out the whole procedure every time, or simply use the general formula we derived to compute the integrating factor (the example in this lecture shows both ways). Either way, you should understand what makes this method work.
    (2) The existence and uniqueness theorem for linear equations is optional (i.e. won't be asked on exams), so is the counterexample when the equation is nonlinear shown at the end of this lecture, but you should get a sense of what they say.
    (3) You don't need to be able to draw the slope field and solution curves exactly by hand for linear equations, but you need to be able to identify some qualitative features about them, e.g. where the slope is horizontal, where the slope is positive or negative, behaviors of the solutions when t approches a singularity and how the different initial conditions affect the behavior, etc.
Notes
L8 Youtube
Panopto 		Textbook reading: 2.3
Keywords: modeling, unit conversions
Modeling example: mixing problems
Notes
L9 Youtube
Panopto 		Textbook reading: 2.5
Keywords: autonomous equations
Modeling example: logistic growth and variants of it
Objectives:
    (1) You should be able to graph the slope field and solution curves by hand for autonomous equations, and classify stability.
    (2) Modeling with autonomous equations.
Notes
L10 Youtube
Panopto 		Textbook reading: 3.1 (optional: 3.2)
Keywords: homogeneous 2nd order linear equations with constant coefficents
Modeling example: harmonic oscillators
Objectives:
    (1) Model for the spring-mass system (meaning of the equation and the initial conditions)
    (2) Form of the set of solutions of a linear homogeneous equation.
    (3) Solutions to a homogeneous 2nd order linear equation with constant coefficients whose characteristic polynomial has two distinct real roots. Once you understand the process, you can write down the solution directly after finding the roots, no need to show more work.
Notes
for L10-12
L11 Youtube
Panopto 		Textbook reading: 3.3
Keywords: homogeneous 2nd order linear equations with constant coefficents
Objectives:
    (1) Solutions to a homogeneous 2nd order linear equation with constant coefficients whose characteristic polynomial has a pair of complex roots .
    (2) Write the solution to (1) in terms of real valued functions. Once you understand the process, you can write down the solution in terms of real valued functions directly after finding the roots, no need to show more work.
L12
25min 		Youtube
Panopto 		Textbook reading: 3.4-3.5
Keywords: solving 2nd order equations iteratively
Objectives:
    (1) Solutions to a homogeneous 2nd order linear equation with constant coefficients whose characteristic polynomial two repeated real roots (i.e. a single root). Once you understand the process, you can write down the solution directly after finding the roots, no need to show more work.
    (2) The iterative method for solving differential equations is entirely optional, but it's helpful to know if you are curious about how to find the solutions to these equations in a systematic way rather than "staring" at it for long enough to intuitively come up with a good guess.
L13 Youtube
Panopto 		Textbook reading: 3.5
Keywords: nonhomogeneous linear equations, method of undetermined coefficents
Objectives:
    (1) Solutions of a nonhomogeneous 2nd order linear equation and how they are related to the solutions of the corresponding homogeneous equation.
    (2) Introducing the method of undetermined coefficients
Notes
for L13, L14(part a)


Notes
for
L14(part b),
L15
L16(part a)


Notes
for
L16(part b)
L14 Youtube
Panopto 		Textbook reading: 3.5, 3.7
Keywards: Method of undetermined coefficients, harmonic oscillators
Objectives:
    (1) Use the method of undetermined coefficients to solve 2nd order linear equations with constant coefficients whose nonhomogeneous parts are of the form (polynomial)*(sin/cos)*(exp), or a superposition of these.
    (2) Two experiments for for finding the spring constant of a spring. Note that experiment 2 is more commonly used (it's easier, also it's more accurate than experiment 1 since it doesn't have as much simplifying assumptions, like being frictionless.)
L15 Youtube
Panopto 		Note: After the lecture, I added a little comment at the bottom of p7 of the notes.
Textbook reading: 3.7
Keywords: harmonic oscillators
Objectives:
    (1) For a point in ℝ2, write in terms of the Cartesian coordinate (c1, c2) and the polar coordinate (A, δ).
    (2) Graph the solution to the harmonic oscillator equation with initial condition, indicating the period, initial conditions, amplitude, and phase shift. Note that the period depends on the intrinsic properties of the system (i.e. spring constant and mass); amplitude and phase shift depend on the initial condition.
L16 Youtube
Panopto 		Note: Since someone asked, I added a comment on the last page of the notes giving a bit more explanation about cosh and sinh.
Textbook reading: 3.7
Keywords: linearization, damping
Modeling example: pendulum, damped harmonic oscillators
Objectives:
    (1) Solve damped harmonic oscillator equation. No need to memorize the formula since these are just homogeneous 2nd order equations with constant coefficients.
    (2) Graph the solutions. Note that the graphs for the critically damped and overdamped cases are similar. In these two cases you don't need to be able to graph by hand precisely. But you should be able to illustrate the key features such as the initial condition, the peak/trough if there is any (i.e. when the derivative=0), the long term behavior (i.e. decay to 0), etc.
L17 Youtube
Panopto 		Textbook reading: 3.8
Keywords: forced vibrations
Objectives for §3.8 (i.e. L17, L18, and the first 8min of L19):
    (1) Solving the equation for forced vibrations. Don't memorize the formulas! Understand how these are simply solved using the method of undetermined coefficients.
    (2) Graphing the solutions. After you finish watching all the lectures on §3.8, I strongly recommend that you read the "gallery of graphs" (linked in L19's extra resources), which provides a summary of what was discussed in the lectures and what you are expected to know.
Notes
for
L17,
L18,
L19(part a)







Notes
for
L19(part b)
L18 Youtube
Panopto

2min correction video:
Youtube
Panopto 		Note!! The beats graphs on Pages 4 and 6 were wrong, sorry! It has been corrected in the notes, and see the 2min correction video explaining this! Hope this will make you pay extra attention to this issue.
Textbook reading: 3.8
Keywords: forced vibrations
L19 Youtube
Panopto 		Textbook reading: 3.8, 6.1
Note: only the first 8 min of this video is on §3.8, and the rest is on section §6.1.
Keywords: forced vibrations, Laplace transform
Extra Resources: Objectives for §6.1:
    (1) Understand the definition of Laplace transform and use the definition to compute Laplace transforms
    (2) It is important to understand that the Laplace transform is only defined on the interval of convergence, though you'll not be required to figure out what that interval is in this class.
Table of Laplace Transforms
Table of Laplace Transforms (without sinh and cosh)
L20 Youtube
Panopto 		Textbook reading: 6.2
Keywords: solving differential equations using Laplace transforms
Objectives:
    (1) Properties 1,2,3
    (2) Solving initial value problems using Laplace transforms for equations whose solutions are continuous.
Notes
Extra brief (9min) review of partial fraction decompositions:
video (Youtube,   Panopto),   notes
L21 Youtube
Panopto 		Textbook reading: 6.3
Keywords: step functions
Objectives:
    (1) Writing a piecewise continuous function in terms of step functions.
    (2) Properties 4,5
    (3) Laplace transforms of step functions and piecewise continuous functions.
Notes
L22 Youtube
Panopto 		Textbook reading: 6.4
Keywords: equations with discontinuous nonhomogenous term (driving force)
 Notes
L23 Youtube
Panopto 		Note: I added a quick exercise at the end of the lecture notes after the video ended.
Textbook reading: 6.5
Keywords: Impulse (Dirac delta) function driving force
Objectives:
    (1) Laplace transform of impulse functions
    (2) Solve initial value problems with impulse functions as the driving force
    (3) Graphing the solution
Notes
L24
8min 		Youtube
Panopto 		Brief comments on the table of Laplace transforms
L25 Youtube
Panopto 		Textbook reading: 6.6
Keywords:convolution
 Notes

Numerical Resources

None are required, and we won't be using them very much in this class, but feel free to play around with any of these as you like.
  • WolframAlpha
  • Geogebra: a nice graphing tool
  • UW IT Connect has a list of softwares you can install on your personal computer, e.g. Mathematica and Matlab are free for students.
  • Python: you can run Python from a browser using iodide, or you can download Python from Anaconda.
  • Sage, which can be run from a web browser using CoCalc. Sage is mostly built on Python but not the same, e.g. it's easier to type math expressions in Sage and and it has more math specific codes/routines built in. Its founder is a former UW math professor, William Stein. Thomas Judson's book, The ODE Project, teaches differential equations and Sage.
  • Slope field and integral curves:
  • Euler's method with Python: eulerExp.py and explanations written by William Casper.