For the first time in a while, I actually had a decent week on KA. I made it through seven videos and one exercise. There were times when I was out of my depth, but for the most part I was able to keep up with and understand what was going on, plus I didn’t feel as rusty doing calculus, algebra, etc. 😮‍💨 Really, I still should have put in a better effort and made more progress—I’m fairly certain I studied less than 5-hours—but I’m just happy that I finished the week not feeling terrible about my effort, like I have over the past few weeks. In general, I still don’t fully see the bigger picture of what’s going on with differential equations, but I’m making progress through the unit nonetheless, so I’ve at least got that going for me which is nice. 🙃

Video 1 – Exponential Models & Differential Equations (Part 1)

This was the first video from a section titled Exponential Models. To be honest, I don’t really understand what makes ‘exponential models’ different from the other differential equations. The good news was that I found the calculus and algebra pretty simple to follow along with and solve. The question was saying something like, what is the differential equation for the rate of change of a population at any given time. As you can see, all that needed to be done was separate the variables, integrate both sides, raise each side of the antiderivative equation to e and then use some algebra to solve for P. Boom. 🧨

Video 2 – Exponential Models & Differential Equations (Part 2)

This vid was a continuation of the first one where Sal just gave some values to plug into the equation. My notes look like a squiggly mess of spaghetti, but if you follow along the arrows from (1), (2), and then (3), you can see that you first solve for C and then K, which are the unknowns, and then plus their values in to get the solution to this differential equation. 

(I have no idea if I said that right or if it’s even correct…)

Video 3 – Worked Example: Exponential Solution to Differential Equation

This video is essentially the exact same as the previous video, but I believe there was no story behind it. I’m pretty sure Sal just said, “let’s solve this differential equation”. Between the screenshot and the photo of my notes, you can see how you first solve for C, then input C back into the antiderivative equation to find the solution to the differential equation.

(Again, no clue if I said that right. Or if it’s actually true/correct.)

Exercise 1 – Differential Equations: Exponential Model Equations

This was the only exercise in this section, and unfortunately the final exercise for the entire unit. (I say unfortunately because I find the exercises SUPER helpful to better understand what’s going on, and even though it delays me getting through this sooner than I would, sometimes the questions can be somewhat fun to solve.) I got all four questions from this exercise correct on my first attempt which I was pumped about. They were solved with the same method that I just described in the last two videos, so I’m not going to bother explaining my notes for these two questions:

Question 1

Question 2

Video 4 – Newton’s Law of Cooling

I still don’t really understand what makes this question an ‘exponential’ differential equation, but it was nice to work through this video to see how differential equations work and get a better idea of what they are. It’s hard to explain (mainly because I don’t completely understand it), but the gist of what was talked about here is that there’s a differential equation you can use to determine how fast an object would cool down or heat up in a room that has a different temperature than the object. So, for example, if you had a cup of hot coffee in a room, the differential equation would tell you how long it would take for the coffee to cool down room temperature. As you can see in my notes, I was able to follow along with the calculus and algebra as Sal was going through it but looking back on my notes now, I understand the gist of what’s going on but don’t have it completely figured out. 🤷🏻‍♂️

Video 5 – Worked Example: Newton’s Law of Cooling

This was the final video from the Exponential Models section and continued on from the previous video. In this vid, Sal took Newton’s law of cooling and plugged some values into it. Like before, I understood the gist of what was going on, but definitely need more practice with these types of questions/equations to get more comfortable with them and a better intuitive grasp of what’s going on. But on the plus side, I worked through this question before watching Sal solve it and got all the calculus and algebra correct. I screwed up on a log rule at the end (not shown in my notes) which was annoying, but overall I was happy that I didn’t get stumped on the calculus. 

Video 6 – Exact Equations Intuition (Proofy)

This video was from a section titled Exact Equations and Integrating Factors and I found it very confusing. It was clearly one of Sal’s OG videos (I went to YouTube and it was uploaded 16 years ago 😳) and he may not have been as good at setting up the premise for the video back then as he is now, or it just as easily could have been that I straight up didn’t understand what was going on… Either way, I watched this video and the following one (which I didn’t make notes on) and then stopped as I didn’t understand what was going on but also thinking that I would start fresh with the entire section this upcoming week.

And that was all I got done this week. Not great, but also not the worst week I’ve ever had. I have 14 videos left to get through in this unit. I think it’s possible for me to get through it all in two weeks, but it will be tough. I would be PUMPED if I could make it happen, especially given how slow I’ve been going lately. It’d be awesome if I could get some momentum going again this upcoming week, so, as always, fingers crossed I can get it going! 🤞🏼