This may have literally been the worst week I’ve ever had on KA. 😒 I probably studied for just under three hours, so that in itself was bad, but I only made it through one exercise and two videos… The exercise kicked my butt until around Friday for a few reasons, but then I started to get a hang of how to solve the questions. The main reason why I was struggling was because I didn’t understand (and still don’t) why differential equations work or what they even are. At the beginning of the week, I also had no clue how to solve the questions but eventually figured out the pattern and got the hang of it. Another reason I was getting rocked was because I’m super rusty with integrating expressions, log rules, and algebra, in general. As frustrating as it was, I was happy to get some practice with all of those things and started to remember it all by the end of the week. Like I’ve said before, I actually enjoy doing this type of math (compared to linear algebra, at least) as it’s kind of like solving little puzzles, but being rusty and screwing up simple algebra was annoying and disappointing. Some other excuses I have for my poor week are 1) I’ve been sick this whole week, 2) my work schedule was different this week and I had to leave for work each day at 11:15am as opposed to ~3pm, and 3) I’ve been spending a lot of time working on the other project I’ve talked about—not that that’s much of an excuse—which has me less concentrated on KA. (I’ll explain/talk about that project soon!) So, all in all, not a great week, but still a bit of progress nonetheless.
(Longest opening paragraph ever.)
Exercise 1 – Particular Solutions to Separable Differential Equations
Question 1
This was the first question I worked through this week and I got it wrong. Worse than that, I still don’t understand why or how the solution works the way it does. When I tried to solve it, you can see that I moved 3y over to the left-side of the equation turning it into 1/(3y). Apparently, I was supposed to leave the 3 over on the right-side of the equation making it 3x0. As far as I understand it, with algebra you should be able to move scalers to either side of the equation by multiplying both sides of the equation by their reciprocal keeping the overall value of the equation the same. So, I either screwed up the algebra after moving the 3 over the left-side, or moving the 3 over just straight up doesn’t work. 🤔 Either way, I got this question wrong and began the week feeling pretty unsure of what I was doing. 😒
Question 2
After a few days of getting questions wrong, I finally got this one correct on Thursday or Friday. By the time I got to this question, I had gotten pretty good at following the pattern of separating the questions’ equation’s x- and y-variables, solving for C, then throwing whatever that constant C represented back into the equations and solving for the unknown x or y variable. (You can see that in this question it was t instead of x.) Once I figured out the pattern, these questions weren’t too difficult, although some of the calculus, log rules, and algebra was tricky at times. I also never got to the point where I intuitively knew what was going on, BUT at least I was able to solve the questions. 🙃
Question 3
As you can see, this follows the the exact same format as the previous one.
Question 4
This question is essentially the same as the previous two but asks you to solve for the coefficient and exponent on e. I’d seen this exact type of question before and previously got it wrong, so this time I knew what the pattern was to solve it. When I got it wrong initially, I couldn’t remember the log rules. I also didn’t understand that the question was stating that the solution to y for f(3) would take form of men, so that was part of my confusion the first time I tried to solve this type of question. Anyways, when I worked through the question above I was happy that I remembered the pattern from before and was able to do the log rules and solve the question fairly quickly.
Question 5
This question is clearly the same as the second and third questions from above, but it still gave me some trouble. It was the algebra at the end that I was stumped on for a while. I couldn’t think through how to solve 1/y3 = 8. I eventually multiplied each side by y3 and 8, at which point it was clear to me that the cubed root of 1/8 was 1/2, so I ended up getting it correct in the end, but it was pretty frustrating not know what was going on and not remembering how to solve some pretty straightforward algebra.
Video 1 – Separable Equations (Old)
This video and the next were the final two vids from this section, Separable Equations. This video was Sal working through two questions, which you can see in my written note above. After working through the exercise, the examples he gave in this vid (and the next one) were very easy, so I’m not going to bother explaining how to solve them here.
Video 2 – Separable Equations Example (Old)
And that was it for this week. 😔 I have 19 videos left to get through in this unit and one exercise. I have a feeling it’s going to take me at least three weeks to get through it all. If that’s my goal, I NEED to be more productive this upcoming week and make WAY more progress than I did this past week. I know I’m a broken record by saying this, but I’m so close to finishing this off and should be pushing myself harder to get there. I know I will eventually do it, but it needs to happen sooner rather than later. 🤬😤