It was a good but also bad week for me on KA. I made it through five videos and three exercises which was pretty good, but I spent only ~3 hours studying, which is pretty bad. I was happy to get back to doing exercises which felt like a throw-back. (I can’t remember the last time I did an exercise, but I’m guessing it was at least three months ago.) One positive was that I was happy with how much I was able to remember from calculus. Of the 15-20 questions I worked through this week in the exercises, I only got a handful of them wrong. That said, my understanding of differential calculus is more or less non-existent. It has to do with equations where the solution is a function. I find this concept hard to wrap my head around, but that seems to be the gist of what differential equations are. Even though I find it a bit daunting and overwhelming right now, I’m somewhat pumped to get further into the course to get a stronger grasp on what Sal is talking about. 💪🏼

Video 1 – Differential Equations Introduction

This was from the first video in the entire course, and when I started watching, I had flashback to when I saw it a long time ago. As you can see in my written notes, Sal explained that a differential equation is an equation where the solution(s) is/are a function or a set of functions. Without doing a full-on proof, he showed in the example that 3e^-3x was a solution to the differential equation on the right-side of the screenshot.

Exercise 1 – Verifying Solutions to Differential Equations

Question 1

To solve this question, I had to find the derivative of the function f(x) = 2(ln(x))³ and then input the function back into the equation f’(x) = 3f(x)/(x(ln(x)) to see if the solution was equal to the derivative of f(x) = 2(ln(x))³. As you can see, it turns out that the derivatives were in fact equal, meaning the function f(x) = 2(ln(x))³ was indeed a solution to f’(x) = 3f(x)/(x(ln(x)).

I followed the same process to solve the following two questions from that exercise:

Question 2

Question 3

Exercise 2 – Write Differential Equations

The following two questions were from the exercise Write Differential Equations and were all about reading a given statement and choosing the correct equation the represented what the statement was saying. I got all four questions correct on the first time but found them all pretty challenging. The wording was generally pretty confusing, and knowing what words denoted what operations was hard to think through.

Question 4

Question 5

Exercise 3 –Differential Equations Challenge

This was the hardest of the three exercises I worked through this week, but I managed to get all four questions correct on my first attempt. However, I don’t think I solved a single question the way Sal did, so I’m not sure if I just got lucky or if I was thinking through the questions in a weird way but still ‘correctly’.

Question 6

I’m not going to try to explain how I managed to get to the correct solution on this question because I honestly don’t know how I did. I remember spending ~10 minutes trying to think through how the question worked, essentially giving up, coming to a solution that I didn’t think made sense but seemed like the only logical way of solving the question, and then I happened to be correct without having any clue as to what I was really doing. 🤷🏻‍♂️

Video 2 – Worked Example: Linear Solution to Differential Equation

This video was essentially the exact same type of question as the one from the previous exercise. I watched this video before doing that exercise which is the only reason why I was able to get to the correct solution from that previous question, otherwise I would have been completely lost. I don’t have much of a grasp of what’s going on here, but I was able to do the algebra and calculus pretty easily, so I’ve at least got that going for me which is nice.

(Also, I didn’t make any notes on the second video from this section, Writing a Differential Equation. It was a ~2-minute video that was a quick example of writing a differential equation.)

Video 3 – Slope Fields Introduction

This was the first video from the following section, Slope Fields. In the vid, Sal gave the equation dy/dx = -x/y and plotted the slope at a few different points on the plane which showed the cyclical nature of slope from the equation.

Video 4 – Worked Example: Equation From Slope Field

This was another video I remembered watching a long time ago. In it, Sal started by talking about the slope field and asked what equation could represent it. Without looking at the equations on the left, I was able to figure out the solution would be dy/dx = (x + y) based on each of the little slope lines and doing some trial-and-error math in my head. To be honest it wasn’t that difficult to figure out, but I was still pretty pumped that I was able to figure it out on my own.

And that was it for this past week. I’m a bit disappointed that I didn’t make more effort to get more done, but I’m happy with the number of videos and exercises I got through nonetheless. There are about 20 videos left in this unit and five exercises. I’m hoping I can get through it all by mid-April. The start of this unit has been fairly straightforward and I’m guessing it will get a lot harder as I get further into it, but I’m optimistic that I’ll be up to the challenge. As I mentioned at the end of my last post, I’m pretty excited to be back in calculus as I find it more enjoyable than linear algebra. I typically find subjects easier to understand when I come back to them too, so fingers crossed that happens again. 🤞🏼 Hopefully it Differential Equations won’t end up being too difficult so I can FINALLY get to the end of the math section in KA sooner rather than later. I know I keep saying this at the end of each post, but it’s been almost SIX YEARS since I started this and I’m ready to finish it off. Not because I want to stop studying math, but just because I want to FINALLY be able to tell myself that I achieved the goal I set for myself five and a half years ago. 😮‍💨🙏🏼