I had another setback this week working through KA. I began this week by getting back into the unit Differential Equations as there were still two sections left in this unit to complete. The first section was titled Approximation with Euler’s Method which contained 2 videos and 1 exercise. I’d already watched the first video and so I assumed it wouldn’t take me long to get through the section. I watched the second video and then tried the exercise but had no idea what was happening or how the questions worked so I decided to go back to the start of the unit and rewatch all the vids. I’m happy to say that by the end of the week I got back to the Euler’s Method exercise, got through it, and now have a decent grasp on how this method works. Even still, I can tell that I have a LONG ways to go before I actually know what’s going on with differential equations but, nonetheless, I feel like I made some serious strides towards understanding these equations which I’m pretty happy about.

After getting rocked by the Euler’s Method exercise, the first thing I did was Google differential equations to see if I could find some other sources that’d help me understand differential equations. One of the first videos I came across was the following video from a creator called 3Blue1Brown:

After watching this video I realized I had (and still have) no clue what’s going on with differential equations, BUT it looks like I’m in a good position having gone through Calc.1 to be able to figure them out. Although the vast majority of the math from this video was over my head, I found this vid very motivating after watching the presenter work through the example of the pendulum. Even though I don’t really know what’s going on, based on that example I can tell that using derivatives and slope fields are an integral part of physics. It’s nice to see a concrete example of where and when calculus plays a key role to solve real world problems which I find pretty inspiring.

After watching that video a few times, I rewatched the first few videos from the unit and made this note:

At the top of the page you can see the different types of notation used for differential equations:

• Differential Equation Notation
  • Leibniz Notation
    • 3*y = 2*dy/dx + d²y/dx²
  • Lagrange Notation
    • 3*f(x) = 2*f’(x) + f’’(x)
  • Cartesian (?) Notation
    • 3*y = 2*y’ + y’’

The bottom half of the note is an example of the fact that there can be more than one solution to a differential equation.

On Wednesday I made it to a section that covered slope fields. I remembered going through these videos and exercises from this section a long time ago but my memory was pretty hazy on how it all worked. Here’s two pages from my notes where I graphed a slope field based on the differential equation dy/dx = –x/y:

The first thing to do when drawing a slope field is to pick any random point on the Cartesian plane that’s graphing the differential equation. In my notes above, the first point I picked was (1, 1). You then input the x- and y-coordinate into the derivative which gives you the slope of the tangent line at that exact point. You then make a little dash on the Cartesian plane indicating the slope of the differential equation at that point. (I have no idea if what I just said makes sense, but that’s how you draw a slope field…)

On Thursday I got back to the section on Euler’s Method which ended up taking me the rest of the week to get through. I was having a hard time understanding what the concept of Euler’s Method does so, after struggling to figure it out, I eventually told myself to first try to figure out how the method works instead of why the method works. What I mean by that is I decided to try to first figure what the steps used in the process of E.M. before trying to understand what the math was indicating. As much as I’d prefer to have a solid grasp on WHY certain mathematical concepts work, I’ve found over the past three years that it most often works the other way around where first I learn HOW to solve the questions and then I figure out WHY it works the way it does. 

I just tried to write out my understanding of E.M. and realized I don’t know what’s going on well enough to begin to attempt to explain it. Instead, here are 3 example questions from the exercise I worked through which will do a better job at explaining how it works than I can:

Question 1

This question gives you the delta-x (i.e. the interval/change-in each x-value) which is 1. To solve this question you start by inputting the first set of (x, y) values, (2, 3), into the differential equation:

• dy/dx = x + 3y
  • = 2 + 3(3)
  • = 11
  • m = 11

This means that the slope of the tangent line at (2, 3) is 11. Knowing that m = 11, you can then find what I think of as y₂ by using the formula m = y₂ – y₁/x₂ – x₁:

• m = y₂ – y₁/x₂ – x₁
  • 11 = y₂ – 3/ 3 – 2
  • 11 = y₂ – 3/1
  • 11 + 3 = y₂ – 3 + 3
  • 14 = y₂

Since the question asks you to get the value up to f(5), you then need to repeat the same process two more times to fill in the remaining values.

Question 2

Question 3

Questions 2 and 3 above follow the same type of pattern as question 1 in that you substitute an initial set of (x, y) values into the derivative to get the slope of the tangent line at those coordinates, then use a pre-set delta-x and the slope to tell you the next y-coordinate in the following set of (x, y) coordinates. Question 3 also throws a curveball in that you’re not given the initial y-coordinate but rather the variable k in its place. You’re instead given the third set of coordinates, (2, 4.5), but you follow the same pattern to determine the value of k.

Now that I’ve decided to review all the videos in Differential Equations (1,180/1,300 M.P.), I expect it’ll take me at least another 2 weeks to get through it. There are 4 sections left in this unit with 17 videos between them, and the final section has an exercise I need to get through, as well. I’m not stressed about how long it’ll take me to get through this unit though and am not worried about how far through Calc.2 I’ll be when September rolls around. As much as I’d like to get through it all quickly, I’m really enjoying learning about differential equations and think it’s WAY more important to have a strong grasp on them before moving forward into Calc.2 than it is to rush through it quickly. My new goal is to get through Calc.2 by 2023 which I definitely think is doable. 🤓💪🏼