Matrix Systems of Differential Equations
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- Опубліковано 8 лип 2024
- This video describes how to write a high-order linear differential equation as a matrix system of first-order differential equations. This is a major step towards solving general ODEs using eigenvalues and eigenvectors, leveraging the full power of linear algebra.
Playlist: • Engineering Math: Diff...
Course Website: faculty.washington.edu/sbrunto...
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This video was produced at the University of Washington
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0:00 Overview
1:54 Introduce New Variables
7:11 Writing as Matrix System of Equations
12:12 Summary and Takeaways
14:51 Eigenvalues of Matrix System are Roots of the Characteristic Polynomial
17:40 Example 3x3 Matrix System of ODEs - Наука та технологія
2:17 "I actually opened up a new pack of markers because I'm so excited about this lecture" hehe the enthusiasm is infectious
I have to admit that without squeaking, I concentrate less
Those last lectures about ODEs are seriously one of the best ones I've seen on UA-cam! Really enjoying it, keep it up :D
You made me love math and calculus which i hated a long time. The different ways of math notations, best explanation and relationship between linear algebra and ODE is just a thing that can make me study math soon.
New pack of markers! Man, I really should buckle down and watch all of these videos as a refresher. I struggled to really comprehend them during my undergrad, and it'd be nice to finally feel like I fully understood.
Haha I love that you drew a heart at the meeting point of linear algebra and diff equations.
Thanks so much for all these presentations - honestly some of the best material on UA-cam, and so brilliantly created. Big fan.
I love when two different areas of math connect to each other as shown here with linear algebra and diff. eq.. So satisfying!
Absolutely Exceptional explanation how linear algebra combines with the calculus to solve differential equation! I am feeling blessed to find this videos on UA-cam! we love you lectures!!!
I like searching good lectures on UA-cam. This series is as best as Strang's Linear algebra!
This is my new favorite series!!!!
Steve was feeling himself in this one 🤣 a mixed of math and stand up comedy. Loved it
You are always so excited!!! I love it!!
pulling an all nighter watching ur videos, absolute treat
Thank you - you’re a phenomenal teacher.
Matrix systems of differential equations……
I’m so thankful for them!
writing a comment down here , this is such a good video along with such enthusiasm shown by our prof .
Amazing. Thank you!
this video is a gem ❤️
awesome video. thank you
Excellent series of lectures on solving higher-order ODEs. I would request you to make a separate video that talks about the geometrical interpretation of the solution. In my opinion, the interpretation is like this; each of the eigenvalues corresponds to the exponential rate of divergence along the eigenvectors of the Jacobian matrix A. So if x = c1 exp(lamb1 t) + c2 exp(lamb2 t), then there is a eigenvector associated with c1 and c2. The solution can be written as u1 exp(lamb1 t) + u2 exp(lamb2 t). This would lead to an eigenvalue problem where u1 and u2 are the eigenvectors of A. The solution x can now be expressed as c1 u1 exp(lamb1 t) + c2 u2 exp(lamb2 t). This solution can be interpreted as how the vector(solution) grows or shrinks along the axis(u1 and u2). The eigenvectors would be the basis of the solution and lambda's would tell us how they grow in those directions(eigenvectors u1 and u2).
what a beautiful picture of math u made ! thank u for the heart
It is a gem.❤
Nice explanations
Very good
thank you so much, ily
i wish i had these videos when i was in my EE program. back then 3blue1brown started to emerge but he couldnt carry me alone there!
"Polly Polynomial"....Linear Algebra. ❤ DiffEq......I love it! 😂
More please
at 18:50, when I do the matrix multiplication with the vector I get an extra x2 by itself without any a coefficients. However, in the equation below there is only a single variable without an a coefficient. where does that go?
can you please explain how do we represent odd powers as physical spring mass systems as addind additional spring and masses is just providing even power linear ode's
Hello Prof. Brunton, thank you very much for your video and your contributions. I wanted to ask if you can cover the mathematical background of the message from AlphaTensor. DeepMind reports that they have developed an AI-based algorithm that accelerates matrix multiplications.
Thank you very much!
what if the equation isn't omogeneous and the coefficients aren't constant but dependant on a variable?
Thank you for your amazing content. I share your videos all the time on social media. I might be wrong but I don't understand what happened to the minus sign of the characteristic polynomial at the end of the video. Other than this, thank you very much for your effort!
-equ=0 is -1*equ=0 --> equ=0/-1 --> equ=0 (and the minus is gone)
how do they film this?
Why is that so hard to find material on systems of differential equations? This video doesn't even have a lot of views.
ask UA-cam?
Every time I see someone teaching on one of these glass panes, I'm always perplexed at how it is being done. Is the video mirrored or is he writing backwards? Also, if it is mirrored, how is he oriented in relation to the class?
If you watch at the pen tip you can see that Dr. Brunton has to write backwards our left to right but his right to left. He is behind the glass. That is my perspective at least.
How does he record these? Like is there a pane of glass between him and the camera that he writes on or what because it was cool but confusing. Also, is he writing mirrored?
He's probably not talented enough to write mirrored and have it look natural. It is probably mirrored video footage. One way you could do it, is by digitally flipping the video. Another way, is to use an optical mirror.
How do you film these videos?
recorded normally and then flipped
I miss when math was this easy…
this is easy???
😔😢
@@nerd2544 Depends on the field you go into.
No this can't be so smooth, something is wrong lol
Does this guy write backwards or something??!
He mirrors the video footage. If you saw him in person, the writing would be backwards from your side of the glass.
Tõõ small
It's surprising that to get cha. poly. we assumed the form of the solution (exp(lambda*t)) while with the matrix method we didn't do any assumptions and arrived at the equivalent form.