I didn’t spend enough time on KA but, overall, it was still a pretty solid week and I’m happy with the progress I made. I made it through five videos on KA, finished one exercise, and started another. That doesn’t sound too bad, except that the videos were all ~3–8 minutes in length, so I definitely should have made it through more. Even still, I feel like I made a lot of progress understanding Euler’s Method, and then got a better grasp on separable differential equations. I think I’m starting to get a better understanding of what differential equations are in general which, in the big picture, is what’s most important. So, although it could have been better—I feel like I say that every week—it also could have been worse.

Exercise 1 – Euler’s Method

Question 1

This is a great example of why working through exercises helps me so much getting concepts to sink in. After watching the two videos in this section, Euler’s Method, I had a vague understanding of what was going on conceptually, but actually working through this exercise made understanding how the math works SO much easier to understand. 

As you can see, the question gives you dy/dx = x + y (a.k.a. the slope of some function (the differential equation I think 🤔) at any given point equals (x + y), and it states the function runs through the point (1, 2). You then throw (1, 2) into dy/dx = x + y, i.e. 1 + 2 = 3. Since the question tells you to take two steps from x = 1 to get to x = 3, you therefore need to redo the same calculation at both of those points. Since you’re going across the x-axis by 1, you then add 3, the slope, to the y-coordinate (i.e. x goes from 1 -> 2 and y goes from 2 -> 2 + 3 = 5). You do the same calculation to find out that the slope at (2, 5) equals 7, and that at x = 3, y = ~12. Boom. 🧨

The following two questions use are solved by more-or-less doing the exact same thing.

Question 2

Question 3

One thing I’ll mention from this question is that, since each step is only 0.4, you have to multiply the slope by 0.4 each time you add it to the previous y-coordinate.

Video 1 – Separable Equations Introduction

This was the first video from the following section, Separable Equations. I remembered a bit of how to solve these equations, multiplying each said by dx and/or dy as necessary along with the variables to get them on either side of the equation. I didn’t (and still don’t) know what was going on or why the math works. Nonetheless, it was still a useful video to watch and I actually felt pretty good about my grasp on separable D.E. given that I don’t really know what’s going on.

Video 2 – Addressing Treating Differentials Algebraically

In this vid, Sal just talked about how multiplying expressions by dx & dy doesn’t technically make sense, but conceptually it serves the purpose of what we need it to do and makes enough sense for it to be useful. (…🤔)

Video 3 – Worked Example: Identifying Separable Equations

This video was only ~5 minutes long and was just Sal working separating x’s and y’s in the four equations to see which equations were in fact separable. My note above does a good job explaining what is required for a differential equation to be separable, so I won’t add anything further here.

Video 4 – Worked Example: Finding a Specific Solution to a Separable Equations

This is where things got tricky. To solve this question (and the questions below), you have to separate the equation, integrate both sides, solve for C, then throw the value of C back into the integrated equation along with either the given x- or y-coordinate to solve for the missing coordinate. After this video, I was very confused but started to get the hang of it with the next video and exercise.

Video 5 – Worked Example: Separable Equation with an Implicit Solution

This video was the same type of question. It was good practice/review of calculus (which is actually kind of fun, at least more so than linear algebra anyways) and helped me understand what was going on with separating the D.E. and then solving for C and the missing variable.

Exercise 2 – Particular Solutions to Separable Differential Equations

Question 4

This was the final exercise I got to this week. I only did this one question, got it wrong, and then decided I’d come back to and restart the exercise at the beginning of next week. Even though I got the question wrong, I was happy with my understanding of what was going on. If you can’t tell, I actually did everything correct but then screwed up the algebra at the very end by not squaring the binomial by itself. 🤬 Even though I messed that part up, I was actually really happy with myself having essentially answered this question correctly right up until the end. I took it as a good sign that I somewhat know what’s going on and that I’ll figure it out hopefully sooner rather than later. 😮‍💨

As a final side note, another thing I did this week was start a video series by a creator named Steve Brunton. I’m pretty sure he’s a math professor in Seattle. He has a playlist titled “Engineering Math: Differential Equations …” which is really well produced. (On YouTube, it looks like the title actually ends with “…” but I’m not sure if it’s just not showing me what the end of the title actually is. 🤷🏻‍♂️) I started the series and have been watching the vids on my way to work. I find them helpful to understand conceptually what’s going on with D.E. and am hoping that this series will make everything I’m working on in KA seem more intuitive. Anyways, I made it through the first two vids this week and half of the third, so I figured I’d throw them up here just for funsies:

And that was it for this past week. Like I said, not an amazing week but it also could have been worse. I have 21 videos and two exercises left to get through in this unit. I’m assuming that the videos/concepts will get more challenging the further I get into the unit, but even still, I’m hoping I can get through them all at least by the end of April. The next two units are fairly short (around 20 videos each with no exercises) so I may end up finishing off this course relatively quickly. If I can get through it by the middle of the summer, I should have no problem finishing off the Multivariable Calculus Course Challenge well ahead of September, i.e. the start of my sixth year on KA. It seems far away but given how far I’ve come, in a certain sense it also feels right around the corner! Still, I’m definitely ready to get there so I can FINALLY say I achieved the goal I set for myself 5.5 YEARS ago. 😳 😬