Notes
This page contains the notes (by topic not necessarily by lecture) for MATH 341 Differential Equations during the Fall 2026 semester. Sections may be added or edited as the course progresses. The notes are intended to be a supplement to and not a substitute for the textbook (Logan 2015). They are also not meant as an alternative to attending lectures. Students are encouraged to read through the relevant section of notes prior to attending lectures.
Section 1 introduces the basic idea of a differential equation with an emphasis on ordinary differential equations (ODEs). These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 2 covers first-order ODEs including slope-fields, separable equations, linear equations, integrating factors, and applications. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 3 covers one-dimensional dynamical system including topics such as equilibrium points, stability, phase-line, and bifurcations. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 4 covers second-order linear constant-coefficient ODEs and applications. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 5 covers the method of variation of parameters for second-order linear ODEs that do not necessarily have constant coefficients. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 6 covers the definition and basic properties of Laplace Transforms. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 7 covers the application of Laplace Transforms to solve ODEs. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 8 covers the convolution property of Laplace Transforms and impulse forces. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 9 covers systems of first-order linear ODEs and applications. Includes a discussion of matrices. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 10 covers the eigenvalue problem for linear systems of ODEs and applications for solving linear systems. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 11 covers phase plane analysis for linear systems of ODEs, nonhomogeneous linear systems, and applications. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 12 introduces phase plane analysis for nonlinear systems of ODEs. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 13 introduces applications of phase plane analysis for nonlinear systems of ODEs. These notes are also viewable as a slide deck presentation: Open slides in full screen
Section 14 introduces numerical methods and computation of solutions for initial value problems for ODEs. These notes are also viewable as a slide deck presentation: Open slides in full screen
Tips on Reading the Notes
These notes are written to be worked through, not read passively like a novel. Most sections mix formal derivations, worked examples, and Python/SymPy verification — treating them as a scroll-and-skim document will leave you with a false sense of understanding. A few concrete habits make a big difference.
Every derivation and worked example in these notes is something you should be able to reproduce independently, not just follow along with. Concretely:
- When you hit a derivation (e.g., completing the square for a Laplace inversion, or deriving the variation-of-parameters formula), stop, cover the page, and try to redo the steps yourself before reading on.
- For every worked example, pause after reading the problem statement and attempt it before looking at the solution. If your answer doesn’t match, find where your reasoning diverged — that’s usually where the real learning happens.
- Treat the “Show the code” blocks the same way: try to predict what the plot or symbolic output will look like before running or reading it. If you have Python available, re-run the code yourself and tweak the parameters (change \(r\), \(K\), initial conditions, etc.) to see how the behavior changes.
A section you’ve only read with your eyes, and never worked with a pencil, is not yet learned — it just feels familiar.
These notes are dense, and full comprehension on a single pass is not the expectation. A productive rhythm is:
- First pass: read for the big picture — what problem is this section solving, and why does it matter? The “Goals” box and section introduction at the top of each page are designed for this.
- Second pass: work through the derivations and examples with pencil and paper, as above.
- Third pass (later, e.g. before an exam or when the next section builds on this one): skim again with the benefit of hindsight. Ideas that seemed opaque the first time are often obvious once you’ve seen where they’re headed.
If a particular derivation or example still doesn’t click after two honest attempts, that’s exactly what office hours, asking questions in class, or a study group are for — flag it and ask.
These notes are a supplement, not a replacement for the textbook (Logan) or for attending lecture:
- Each notes page states which section(s) of Logan it corresponds to — read the analogous textbook section alongside it. Seeing the same idea presented in two different voices, with different examples, is one of the most effective ways to solidify understanding.
- Skim the relevant section before lecture, so that lecture is reinforcing material you’ve already met once rather than being your first exposure to it. You don’t need to master it beforehand — just show up having seen the vocabulary and the main idea once.
- After lecture, revisit the notes to fill in any gaps between what was covered in class and what’s on the page (the notes may be more detailed or complete than what fits in class time).
- Use the Goals box. Each section opens with a short list of goals. Read it first, and check back against it when you finish. Ask yourself, “can I actually do each item on the list?”
- Don’t skip the “why,” even when you want the formula. Notice how often a technique is motivated before it’s derived (e.g., why Laplace transforms are useful before how to compute one). Understanding the motivation makes the formula far easier to retain and to apply.
- Use the embedded code as a sandbox. The Python/SymPy blocks are not just for display — they’re meant to be run, modified, and used to check your own hand computations. If SymPy’s answer disagrees with yours, that’s valuable information.
- Watch the linked videos when a topic isn’t landing. A different presentation of the same material (different notation, different pace) often resolves confusion that rereading the same page won’t.
- Budget real time. These are not five-minute reads. An approximate time range is posted near the top of each notes page — use it to plan your study sessions rather than skimming a 45-minute section in 10 minutes and assuming you’ve got it.
- The “Summary” and “Key Takeaways” boxes are a self-test, not a substitute. After finishing a section, try to reconstruct the summary table from memory before rereading it. If you can’t, that’s a sign to go back through the section rather than move on.
Access Jupyter Notebooks
Each section of notes has an associated Jupyter notebook, which can be run via Google Colab. Just click the “Open in Colab” button below for the notebook you want. The notebooks contain all of the code used in the notes, and can be modified and can be modified for exploring on your own.
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