## Course description and instructor information
- [**Course Description**](https://catalog.mines.edu/coursesaz/math/): Classical techniques for first and higher order equations and systems of equations. Laplace transforms. Phase-plane and stability analysis of non-linear equations and systems. Applications from physics, mechanics, electrical engineering, and environmental sciences.
- **Prerequisites**: Grade of C- or better in MATH112 or MATH122.
- **Corequisites**: CSCI128 or CSCI102. 3 hours lecture; 3 semester hours.
- **Instructor Information**: Scott A. Strong, PhD,
[email protected], SH205
- I am comfortable with basically any combination of Scott and/or Strong, with or without titles of doctor or professor.
- I will hold **open** office hours from 1pm-3pm on Tuesday
- I will hold additional office hours **by appointment** from 3pm-3:50pm, MWF in SH205.
- https://outlook.office365.com/owa/calendar/
[email protected]/bookings/
- The following will link to a public file structure shared by dropbox
- https://www.dropbox.com/scl/fo/4dzazmu81wt3iog7bw1mz/AHesY0eE7BmE0Tk3Gpt3niY?rlkey=9ss6lrpo2iphl98t05t992ey7&dl=0
- The goal is to mirror this to Canvas, but that will depend on whether I can automate the process.
## First-Order Ordinary Differential Equations
### 1.1: Definitions, terminology, theory, and qualities inferred from the geometry defined by slope fields and phase lines
- The following link takes you to our polling slides
- [[MATH225S25 - Mentimeter Slides]]
- The following dialog links out to videos. All videos can be found at
- [[(Part 1-Videos) Definitions, terminology, theoretical questions, geometry, solution techniques, and models.]]
- [Part 1.1:Definitions, terminology, theoretical questions, Geometry of slope fields, autonomous evolutions, equilibrium points, and phase line analysis **(youtube playlist)**](https://www.youtube.com/playlist?list=PLfzyv9XksyEfMpgAQ60_35QaYzjAHFt2M)
To start the course we introduce the [**definition**](https://youtu.be/pxSTCU106QEof) a differential equation, and [**terminology to classify**](https://youtu.be/qIB6wCZZWj4) them, e.g., order, linear v. nonlinear, autonomous, homogeneous. After this we consider when we might expect to have [**unique solutions**](https://youtu.be/xHgBLIXkBHo) to [**initial value problems**](https://youtu.be/trVE7bRwxps) so that we can know when solution curves are guaranteed to not cross, i.e., solution curves maintain separation wherever the hypotheses of the existence and uniqueness theorem are met.
After this, we consider the [**geometry**](https://youtu.be/LfcBnSs4LVQ) a first-order initial value problem defines, i.e., [**slope field analysis**](https://youtu.be/pz_UWwethvo) [**of solution curves**](https://youtu.be/pz_UWwethvo). Here, we recognize that this [**differential equation defines the slope of an associated solution curve**](https://youtu.be/n4AYqMttDek) at a given point. The solution curve then follows these slopes into the future. If the [**differential equation is autonomous**](https://youtu.be/ilexNqdYwTM), then a single strip of the slope field is all that is needed to recreate the rest and we represent this single strip as [**phase line**](https://youtu.be/d-BJ4BB8CQk). Constant solutions define **equilibrium points on the phase line** and separates the plane into different regions where solutions are either increasing or decreasing. We then find that equilibrium points will either draw all neighboring solutions in, i.e., [**stable sink**](https://youtu.be/VV_3MObvilc), repel them away, i.e., [**unstable source**](https://youtu.be/VV_3MObvilc), or do both, i.e., [**semi-stable node**](https://youtu.be/53MnvsaCaEY).
### 1.2: The solution techniques of separation of variables, the integrating factor method, the method of undetermined coefficients, and Bernoulli transformation
The following dialog links out to videos. All videos can be found at
- [[(Part 1-Videos) Definitions, terminology, theoretical questions, geometry, solution techniques, and models.]]
- [Part 1.2: The solution techniques of separation of variables, the integrating factor method, the method of undetermined coefficients, and Bernoulli](https://www.youtube.com/playlist?list=PLfzyv9XksyEdlBgDTh_YD5L-RfxrZqTwJ)
At this point, we might like to know about the **quantitative details** associated with non-equilibrium solutions, or solutions for more general equations, as opposed to the qualitative phase line analysis. For this we will have to develop procedures to define functions that solve the corresponding ODE. In this class we typically develop four and a learning outcome is to present students with differential equations and ask them to choose and apply a method to determine its solution. **Successful students will be comfortable with the terminology given above, which can be reviewed before or after learning the method.**
[**Separation of variables**](https://youtu.be/n_Op4fAMYMo) can be used if $y' = f(t,y) = g(t) h(y)$, i.e., the [**rule factorizes over the variables**](https://youtu.be/PAEeoPoUxUU), but leads to a [**solution by quadrature/integration**](https://youtu.be/rW9sA4uIbTg). Often the solutions are implicit and when initial conditions are present, we might have to remember that only one of the corresponding explicit solutions may satisfy the initial value problem, [**when multiple are defined as solutions to the ODE**](https://youtu.be/Uf-14XvgFhk).
**If the ODE is linear**, $y' + p(t) y = q(t)$ then it turns out it can be [**brought into correspondence with the product rule**](https://youtu.be/wNbrnPpgwWw) leading to a general solution represented with nested integrals. These nested integrals are not optimal but the [**integrating factor method**](https://youtu.be/BwmIAEnaWZ4) shows us that [**solutions to**](https://youtu.be/EdxeXGQcI0o) [**linear equations have significant structure**](https://youtu.be/EdxeXGQcI0o) that can be exploited, i.e., [**a general homogeneous solution added to a particular solution to the inhomogeneous problem**](https://youtu.be/cI5bOJjnL2I), so that the particular solutions to linear ODE of the form $y' + a y = f(t)$, where