Differential Equations: Lecture 4.1 Preliminary Theory - Linear Equations

This is a real classroom lecture on Differential Equations.

The beginning of the lecture focuses on using the definition of linear dependence to determine whether functions are linearly independent or dependent. I did four examples.

After that we talked about the Wronskian and how it's related to linear independence. Several examples are done in this video.

Finally at the end we talk about the theory of linear differential equations. There is a lot of mathematics in this video. I hope this helps someone.

*Fun Fact*: There is a voice in the background of the video. Why? Well the AC was BROKEN and it was horribly hot(at least in the lecture where I am wearing the red shirt) and so the door was open. So the voice is some other professor in another class. The AC has fortunately been fixed. Note it was so hot the camera overheated multiple times that day!!!

[mohammedabdul]

The world needs more professors with passion like you, I really enjoy the enthusiasm, never change

[yousafhassan]

Does not feel like a professor is teaching feels like a Bro is teaching, Really enjoy your class have bought couple of ur courses on udemy love them as well as

[JohnFL]

Making a difference, one homogeneous linear differential equation at a time. Thanks for sharing these, they really help! 🐴

[TinyMaths]

Prof really reminds me of my statistics and calc II teacher at UNI; same kind of pure enthusiasm for teaching his subject, and had all these great stories, some hilarious; was an absolute joy to be in his classes. He wasn't a professor at the college but was teaching part time. He spent his other time as a rock musician - what a character he was. He used to award prizes too; you'd get a chocolate sweet if you could catch him making a basic arithmetic mistake. Both of these math guys stoked the flames of my passion for math.

[justinwilliam]

Wish we had more professors and people in general like you in this world. You're videos are tremendous help! Thanks for Linear Algebra tip for UCF

[asandalakaje]

you have no idea how your videos really help me, so thank you so much.

[ghost_rip]

Your videos really help me understand these concepts very much! Is there any reason on why you didn't cover section 4.2 ?

[chrisdesrochers]

It seemed more intuitive to me to solve for a and b s.t. f(x1) =a(f(x2)) + b(f(x3)). So for the last set f(x1) = x, f(x2)= x-1 and f(x3) = x+9, then 1x+0x^0 = (a+b)x + (-a+9b)x^0, so we solve for a + b = 1 and -a +9b = 0, where b =1/10 and a = 9/10. So, then these are linearly dependent.

Further, the previous example show a contradiction when solving for a and b, for 0x^0 + 1x + 0x^2 = a(7+x) +b(x^2) we notice that we get the contradiction from 0x^0 = 7a(x^0) or a=0 AND 1x = ax, here indicating that a=1, since a can't be both 1 and 0, then it is impossible to create a linear combination such that f(x1) =a(f(x2)) + b(f(x3)) so the functions are linearly independent.

I'm in a first semester ODE class and I appreciate you're taking the time to show how linear dependence and independence relates to the content explicitly in a concrete fashion. I am having trouble intuitively understanding just what I'm doing in the class and while in the past I might have been happy with a grade as I approach higher level topics I sense that my ceiling for what I can do will depend much more on understanding the qualitative analysis of the problems when they have applied purposes. So, again thanks.

I just glanced at your linear algebra content. My understanding of linear algebra is near zero. Yes, I learned the procedures (and forgot them), but I didn't develop a means to "see" what I was doing. Hope that your content might help there. I see your emphasis on some proofs, so perhaps it might help bridge a portion of my lack of qualitative sense for linear algebra.

If you receive it, God bless you sir.

[ChristobalStacks]

LETS GO LUIGII IDK WHO U ARE BUT YAAAA

[mathiasjeppesen]

Hey Math Sorcerer. Is it possible to get all the other lectures between lecture 3.2 and lecture 4.1?

[mallninja]

oh man I *wish* the first day of each of my classes included a review of the notation...I'm a capstone (and 1/2 of differential equations) away from graduation and discrete math was literally the only class that covered it. The rest just assumed you remembered and/or would figure it out.

[jonko]

Great teaching. Love this but a small correction: At 1:28:30. the word should be 'hence' not 'whence'. Whence means 'from where' whereas hence means 'as a consequence' or 'for this reason'.

[bullseyebill]

@TheMathSorcerer In the first remark at 9:09, why do we can "at least one?" Can every single one of them not be expressed in terms of the others?

Edit: Oh okay, the answer to my question is at 24:55

[sabelmemes]

Doctor is there a possible way to share the notes with us here :D ?

Thanks a lot.

[salsspar]

does using the identity that sin^2(x) = 1-cos^2(x) here work has well. I get c1=1, c2=-3, c3=-2 and when i set x = 0, it works. and i believe it works for all x. So it fits the theorem i think.

[Kishimita]

Could you do a video on why or how wronskian figured that out?

[КирилИванов-ег]

When you were giving the theory for the nth order linear DE around the end of the video you wrote down that there is always at least one fundamental set.Does that mean that if we group all derivatives in a matrix and all a's in a vector there would always be a nullspace producing 0 other than the zero vector or you mean that because the zero vector exist there is always it as a last resort answer.

[HeartsCherryBlossoms]

Which textbook are you using? Is it "A First Course in Differential Equations with Modeling Applications IIE" by Dennis G. Zill?

[Tawarth]

I laughed so hard when you saw that dude sleeping, said wake up, then found out they only had 3 hours of sleep and then said you can go back to sleep.

[EngineermanCullen]

I’m a little confused at what you meant at time stamp 18:16 . Where did the negative come from?