It was a pretty mid week for me on KA, but that’s better than it’s been for me in a while, so it could have been worse. I made it through seven videos which isn’t too bad. They were all pretty short though, so I probably should have gotten more done. The math I worked through was confusing and I didn’t really understand what was going on. I once again was able to follow along with Sal as he worked through the algebra, calculus, linear algebra, and trig, but I didn’t really know what was happening with the big picture of everything. (And by everything, I mean differential equations in general.) I’m still having a hard time grasping the idea that the solution to a differential equation is an equation. But I suppose the fact that I’m able to follow along with how the math is done is a good sign that I’ll (hopefully) being able to figure out what the F is going eventually and why the math works the way it does. One day! 😤
Video 1 – 2nd Order Linear Homogenous Differential Equations 1
This was the first video from the section Linear Homogenous Equations, which was the first section in this new unit, Second Order Linear Equations. There were four videos in this section and, as you can see from the titles of each video below, they were related/all led into each other. This first video explained what a 2nd OLHDE is and that if you have a solution to a 2nd OLHDE, you can scale it and add another 2nd OLHDE solution to it and they will still be correct.
(See, this is what I’m talking about… I have no idea what I just wrote or if any of this makes sense… 😔 I can do the math, but I have no clue what’s going on. 😒)
Video 2 – 2nd Order Linear Homogenous Differential Equations 2
In this second video, Sal starts to work through an example of a 2nd OLHDE. He explains and derives what’s called the Characteristic Equation (CE), which was pretty straightforward for me to follow along with. After you get the CE, you simply factor it and find the roots of r, putting them back into the general solution equation which is y(x) = C1er_1(x) + C2er_2(x). And that’s it… Not difficult to do, but I don’t know what it means…
Video 3 – 2nd Order Linear Homogenous Differential Equations 3
This video was a continuation from the previous one. Once you get the values of r from the characteristic equation, if you’re given values of x, a, and b in y(x) = a and y’(x) = b, then you can input those values back into the general solution equation to find the particular solution for this DE. As you can see, you also have to use linear algebra at the end, but it’s very straightforward.
Video 4 – 2nd Order Linear Homogenous Differential Equations 4
This was the final video in the section and just another example of finding a solution to this 2nd OLHDE, but the values of r were fractions so it was a bit trickier. Factoring the CE wasn’t easy so Sal used the Quadratic Equation to solve the values of r. After that, you just follow the same pattern of using LA and then inputting the given values of y(x) = a and y’(x) = b into the general solution to find the particular solution.
Video 5 – Complex Roots of the Characteristic Equation 1
This was the first video from the following section titled Complex and Repeated Roots of Characteristic Equation. It goes through literally the exact same thing as the previous four videos, finding the solution to a 2ndOLHDE, but explains what to do when the quadratic equation ends up being negative under the radical, a.k.a. when the values of r are complex/imaginary numbers. This video was the first half of the explanation and in it Sal shows how you have to use Euler’s Formula. I remember learning about Euler’s identity and formula literally years ago and being stunned at how crazy it is, but I don’t think I’ve seen it since then, so it was cool to use it here to simplify this solution. Anyways, you can see in my notes how you substitute Euler’s Formula into the general solution when r ends up being complex. In the next video Sal finishes off the derivation for the general equation when r is complex:
Video 6 – Complex Roots of the Characteristic Equation 2
You can see that once you substitute EF into the general solution, you need to use the trig identities cos(-θ) = cos(θ) and sin(-θ) = -sin(θ) to simplify. After a few more steps, it turns out the general solution when r is imaginary/complex is pretty ‘simple’ in terms of the inputs, although I have no clue what it means and could never get to the solution on my own… ☹️
In the end, Sal worked through a very quick example of how it works, and, as you can see, the math isn’t that difficult when you’re just inputting values into the equations.
Video 7 – Complex Roots of the Characteristic Equation 3
In this final video, Sal worked through another example of a 2nd OLHDE that has complex values for r. As you can see, getting to the solution was pretty involved, but since the givens were y(0) = 1 and y’(0) = 0, all of the sin(x) values were equal to 0 so the math wasn’t actually that difficult.
One thing I’ll say is that even though I don’t understand what’s going on or why 2nd OLHDE work the way they do (or DE’s for that matter), I think it’s pretty sick that I have to use algebra, trig, calculus, and LA to solve these problems and that I actually understand how all of those branches of math work. So I’ve at least got that going for me which is nice. 🙃
And that was it for this week. I’m not going to lie, this was a pretty frustrating post to write… Writing it made me realize just how lost I am with everything. Like I just mentioned, I’m happy that I understand how the math works but definitely frustrated overall. At this point I really just want to get through these next two units, finish the MVC course challenge, and move onto physics. I’m pretty sure Sal said in the very first video of this unit that these 2ndOLHDE’s are used a lot in physics, so even though I’m pretty lost right now, maybe when I come back to them in physics they’ll start to make a bit more sense. I think that actually makes sense since DE’s all seem a bit abstract right now but if I were to apply them to some hypothetical real-world problem, it might ground them in reality and they might make a bit more sense. Either way, I have six videos left in this unit so I’m thinking I should be able to get through it by the end of next week. As usual, fingers crossed! 🤞🏼