How to Take Notes in Differential Equations: A Student's Complete Guide

Differential equations is the math course where choosing the right method matters more than executing it. Your professor presents an equation, spends five minutes explaining why it is classified as a second-order linear homogeneous ODE with constant coefficients, selects the characteristic equation approach, and solves it in three lines. But if you misidentified the equation type — maybe you thought it was a Bernoulli equation or missed that the coefficients were constant — you would spend thirty minutes applying the wrong method and get nowhere.

The classification step is almost always verbal. The professor looks at the equation and says "this is separable because we can factor the right side as a product of a function of x and a function of y." That verbal reasoning is the entire key to the problem, yet traditional notes only capture the mechanical solution steps that follow. Missing the classification logic means your notes show you how to solve this specific equation but not how to recognize similar equations on the exam.

Solution methods also look deceptively similar. Integrating factors, variation of parameters, undetermined coefficients, and Laplace transforms can all solve the same types of equations, but each has situations where it works best. Your professor explains these preferences verbally, often with comments like "you could use variation of parameters here, but undetermined coefficients is much faster because the forcing function is a polynomial." That nuance disappears from handwritten notes that only show the method that was actually used.

Differential equations requires notes that capture both the classification reasoning and the solution procedure. Here are five strategies:

1. Write the classification before the solution for every example. Before your professor begins solving, write a header that identifies the equation type: "2nd order, linear, homogeneous, constant coefficients" or "1st order, separable" or "exact equation (check: My = Nx)." This classification label is the single most important piece of information for each example because it tells you which method to use. When reviewing for exams, you can scan your notes by classification type and see all examples of each equation type grouped together, which builds the pattern recognition that exam problems demand.
2. Note the professor's decision tree for method selection. Differential equations professors have a mental flowchart: Is it separable? If not, is it linear? If linear, is it homogeneous or nonhomogeneous? What order? What type of coefficients? What type of forcing function? When your professor walks through this decision process verbally, capture it as a flowchart or decision tree. This single diagram — which most textbooks bury across multiple chapters — is worth more than fifty solved examples because it teaches you how to approach any new equation.
3. Annotate each solution step with the mathematical justification. Don't just copy "multiply both sides by e^(integral of P(x) dx)" — write why: "integrating factor chosen so that the left side becomes d/dx [mu(x) * y]." When the professor says "this step works because of the product rule," note that. These justifications turn mechanical procedures into understood techniques, and they are exactly what professors test when they modify a problem slightly from the standard form.
4. Create a method summary page organized by equation type. After each lecture, add to a running summary: equation type, identification criteria, solution method, and one worked example reference. By the end of the course, this single page is your exam cheat sheet (for courses that allow one) or your study guide. The act of organizing methods by equation type forces you to understand the structure of the course, which is the meta-skill that separates students who can solve unfamiliar problems from those who can only reproduce examples they have seen before.
5. Record the professor's classification reasoning and search by equation type later. The most valuable content in a differential equations lecture is the two minutes the professor spends explaining why a particular equation is classified a certain way and why a particular method is chosen. Recording with Notella captures this reasoning. When homework presents an equation you cannot classify, search your transcripts for the equation type and find the professor's full explanation of the identification criteria and method selection logic.

The biggest struggle in differential equations is not the algebra — it is the classification. AI recording captures the professor's verbal reasoning about why an equation belongs to a particular type and why a specific solution method is appropriate. This reasoning, which happens in the first two minutes of each example, is precisely what traditional notes miss.

With Notella, you can search "Laplace transform" and find every lecture segment where the professor used or discussed this method — when it is preferable, what types of forcing functions make it the natural choice, and how to handle initial conditions. You assemble a complete picture of one method across all the lectures where it appeared, rather than trying to piece it together from examples scattered across your notebook.

When working on homework, the ability to search your lecture transcripts for specific equation types is transformative. "Bernoulli equation" pulls up the professor's explanation of the substitution that linearizes it. "Variation of parameters" retrieves the general formula and the professor's explanation of when to use it over undetermined coefficients. This targeted retrieval turns your recorded lectures into a personalized reference that is far more useful than a textbook.

Differential equations rewards students who focus on method selection as much as method execution. Here is the workflow:

Before lecture: Read the textbook section to know which equation types and solution methods will be covered. Understanding the terminology means you can follow the professor's classification reasoning in real time.

During lecture: Record with Notella. Write the equation classification before each solution. Capture the professor's decision tree for method selection. Annotate solution steps with justifications. Let the recording handle the detailed algebra you cannot copy at full speed.

After lecture: Review the Notella transcript to fill in classification reasoning and method selection logic. Update your running method summary page. Generate flashcards that present equation types and ask for the appropriate solution method. Work through homework problems using your transcript as a reference for the classification and method selection steps.

This approach builds the problem-solving judgment that differential equations exams actually test — not just whether you can execute a method, but whether you can choose the right one.

Stop choosing between understanding and writing. Record your next Differential Equations lecture with Notella. Try Notella Free and see the difference.