The 18.03 (DiffEq) ASE Experience

One of the requirements for obtaining a course 18 (math) major is differential equations. We have a standard diffeq class (18.03) which almost everybody at this school takes because understandably, it's an engineering school, and one needs to understand how to model things that change over time in their engineering lifetime. There are also other classes that fulfill this requirement, like 18.032, a more theoretical approach to the 18.03; 18.152 P(artial)DEs; or 18.303 L(inear)PDEs. I think it's interesting to note the differing tenths place. In the math department, the tenths place of our decimal course listing denotes the field that it's in. 0 are intro, general classes that many non-math majors take, 1 is reserved for our analysis department, and 3 is continuous applied mathematics (as opposed to 4, which is discrete). Course numbers are fun and all, but a tangent for another time.

Anyways, I have no plans of doing anything related to engineering after getting deeply humbled by 8.01 Mechanics. Well, not deeply humbled, I just found it hard to comprehend anything pertaining to the physical realm (classic math major) but was able to regurgitate enough integrals to get myself a pass. As a self-proclaimed algebraist, I am genuinely not going to need to deal with differential equations in my mathematical career (hopefully), and so I decided to take the ASE (Advanced Standing Exam) at the start of my sophomore year. I wanted to get the credit for the class without spending time sitting in a lecture that I know I won't pay any attention to because my ego tells me that I'm above introductory math classes.

Well, as you all know, I had a lot of traveling and PROMYS in my summer, so I didn't really touch diffeq until after I got back from all that. From then, I just had two weeks before the exam. Yippee.

Before we take the exam, we have to turn in a giant homework packet. It's really helpful, because not only are there problems, there are also suggested readings for each unit, which gives me a very handy plan for studying all the content. The homework is split into two parts. The problems in the first part had solutions and were more standard questions. The second part did not have solutions and involved one to apply the concepts a little deeper than in the first part.

I've never really self-studied anything before and the task felt really daunting. Like, my big ego and I signed myself up to learn an entire semester's worth of mathematics (well, it can't be that much because it's an 18.0 class, SHUT UP EGO!!! Though I will say that many of my non-math major friends also think that 18.03 is not a difficult class) in two weeks. On top of that, I don't really have much motivation to do it because it turns out that motivation comes hard when you have a Valorant addiction.

A word of advice for people who are trying to self-study something: the beginning is really hard. Once you get the ball rolling, the ball just rolls for you. Progress is such a weird thing to track, truly. You're hyperfocused on it at the beginning, but once you get into the rhythm of things, you don't realize the crazy amount of progress you've made. You gotta just dedicate a little bit of time everyday, say an hour, where you're forcing yourself to sit at your desk and work on the subject. It doesn't really matter how much you do, and pretty soon you'll find that you'll want to work on it for longer and longer. It's odd really.

And so get the ball rolling I did. I spent probably 1-4 hours on it on any given day, averaging at 2. Every morning, I opened the same three tabs (18.03ase-hw-part1.pdf, 18.03 Supplemental Notes, and 18.03 Homework - Online LaTeX Editor Overleaf). I took notes in a folder I bought in China as I read; it was very cute. I worked on the associated problems in the first part. I didn't know how to solve them most of the time, because the notes were highkey bad, so I had to look at the solutions to solve the problem. I found my lack of understanding only mildly concerning. "It's fine," I told myself, "I'll figure it out in the second part." Two or three days before I had to leave for campus, I finished the first part of the homework.

The second part had one or two problem per section but no answers. I ended up talking about the second part extensively with my grandbig (<3) from my sorority about these problems. It was fine for the most part. I used ChatGPT to check my work on some of the questions. It was a good review and application of the concepts and genuinely solidified my understanding. There were a lot of assignments that required use of Mathlets, online interactive simulations which gives you good visual intuition of the concepts. It's on the internet, so I recommend looking at it if you're also learning differential equations. That was probably the most helpful, especially since the subject itself involves a lot of graphs. It gave me a lot of intuitions for the concepts!

I had to do a lot of it on campus in between all the other stuff that I was doing in orientation week (sorority recruitment prep, hosting REX events). I was decently productive through all of that, though, and worked through the remaining problems at a decent pace. On the day before the test, I only had one problem left. That wasn't the most ideal spot to be in because I wanted to have the time to do some high-level review of all the concepts so that they're all organized in my head.

A quick tangent about how I study as a math major: doing practice tests is not the best way for me to learn. So many study gurus online will tell you that practice tests will help you so much, which is true when your tests are straightforward. However, in my experience, the tests I've done, especially the ones in pure math, aren't formulaic at all. You have to synthesize a solution, and that comes from first having a rigorous understanding of the concepts, and then applying that knowledge in creating a solution.

That's why when I review for tests, the most important thing that I complete is compiling a review sheet for all the information that was learned throughout the semester. Usually, I am able to put summarize all that information in a couple (like 4-6) pages. I imagine myself giving these notes to someone else, and so I make sure to explain all the concepts as simply as I can. It's a form of the Feynman method, which has been my go-to study method since my senior year of high school. After doing that, I often find myself thinking: "Wow, it doesn't feel like I've learned a lot this semester." It genuinely doesn't feel like a lot and makes the test feel so much more manageable.

Anyways, back to the story.

I finished and printed the homework at 10pm. I started working on my review shortly after that. I started off by writing a list of the main concepts of all the things that was taught in a short outline. Then, I set off to fill in all the details, writing down all the definitions and formulas necessary. Differential equations was green, I decided. I wrote section headlines in a thin green ClickArt marker, and highlighted formulas with my light-green Mildliner.

I don't remember exactly how long it took, but it was something like 4-5 hours, which isn't a lot compared to some of my other classes. I suppose it is true that it's not a difficult class to self-learn (certainly compared to major-specific classes, it shouldn't be harder). The class really comes down to learning a bunch of techniques for solving certain types of equations, and some theory about what certain words mean. After creating my study sheet, I felt quite prepared for the test, and so I hit the hay.

I showed up to the testing room the next morning and saw a few familiar faces. I said hi to my friends and sat down in a chair in the front row. I read through my study guide and reviewed some formulas one final time.

The test came, and it was chill. The questions were very reasonable. I got stuck on a problem for a long while because I forgot that i³ is -i. I figured it out eventually. I had plenty of time to daydream. For about ten minutes, after I had given my initial solutions to all the questions, I closed my eyes and rested my heads for like ten minutes.

Fast forward a couple weeks and I hear back from the registrar that I passed the class. I suppose I knew that much, at least. They didn't tell me my grade, and I asked my advisor about it. He said that they probably don't keep track of that information because it just shows up as P/F on your transcript. This is not what I heard about how testing out of a class works. I don't particularly care either way, so I'll just let things run its course. x