Laplace Transforms and Differential Equations

[@NowanIlfideme]

I can't get over the fact that these lectures really excite me in ways many of my professors didn't. :)

[@wangminghui2810]

Sure, same feeling :)

[@DiamondSane]

Repeating is the mother of learning. And recognition is why you got this excitement.

[@aaronong8358]

I'm always exhausted from work, and my work isn't exactly related to his lectures, but it's so entertaining and interesting to learn from Steve! Great lecture!

[@黃祐宇]

such a nerd

[@JonOlaivar]

I'm not sure that words can describe the positive impact you've had on peoples math education. Such great lectures, I really appreciate you putting the time and effort into them.

[@xiaowenzhang8631]

I just spent 2 days binge watching this series of videos, they are so intuitive and clear and I've learned a ton. Thank you Prof. Brunton!!

[@jha7127]

Best math lectures on youtube i have ever seen.

[@House19881]

amazing! I've never seen a better summary of a Laplace ODE physical example, with the final glimpse of how control theory gets in the way...

[@rajendramisir3530]

Indeed! Extraordinarily cool. I really enjoyed your crystal clear approach to and explanation of solving this spring motion dynamics ODE. I like the control force part and would like to see how you handle it. I got a good grasp of the Laplace/Inverse Transform and used them to solve ODEs in my Differential Equations class taught by Professor Leif at KCC(CUNY). I recall using a lot of exponential and trigonometric identities to solve those ODEs. Laplace was a clever and intelligent French Mathematician - one of my favorites.

[@vnickleswitter]

Thank you for taking the time to discuss these topics. I appreciate it.

[@BentHestad]

An excellent, little lecture! Could hardly been done better. Thank you, Sir!

[@danielhoven570]

This brings back memories from sophomore year :) Thanks for the excellent content professor!

[@PetrGladkikh]

4:59 - I think itwas necessary to mention that laplace transform has linear properties and that is why one can apply it to each term separately.

[@andinosa]

I loved your videos on the Laplace transform. Your explanations is simple and to the point. However, I want to mention for the enthusiasts out there: except for very simple ODEs (e.g.nlinear with constant coefficients) you never use the Laplace transform to solve an ODE. For real life problems you use numerical methods like RK4. Numerical methods can easily solve any ODE, however complex. They should be taught above all else in engineerinf. Unfortunately many old school professors still waste student time with strategies to solve ODEs analytically.... Conceptually interesting but useless...

[@wiloberlies9598]

Great presentation again Dr Brunton, thank you for making this available!

[@toraibo5919]

As I see your lecture, I want to become a professor one day :) Incredible videos, you really helped me out. Greetings from Germany.

[@alexli4472]

Wonderful lecture! I am really looking forward to see the lecture about solving PDE with Laplace transform.

[@pengfeixianyu2268]

Thank you Brunton for these wonderful videos.

[@Unidentifying]

[Gordon Ramsey voice] beautiful . Now just a drizzle of olive oil

[@wangminghui2810]

Why are u sure it's a drop of olive oil? :P

[@valor36az]

Fantastic format and explanation

[@maxmalakjan6137]

You mentioned in the first and second video on Laplace that you can turn a pde into an ode, how do you do that

[@sachinr3823]

Sir, thanks for the lectures. I am following your DMD book exercises problems on flow past cylinder. I want to try these methods on new data,(jets, plumes data). its request for you can please mention some PIV or other image data sets. Thanks

[@sachinr3823]

Sir, please recommend some PIV data repository for jets, plumes

[@LucasDimoveo]

I apologize for the non-mathematical comment, but I must ask - are you writing backward on a transparent board? If so ... that is amazing. Simply skillful. Thank you for your awesome work.

[@mohab.ahmed.033]

No Professor Steve is left-handed, I guess he shoots the video reversed then flips it to normal in editing

[@MrAri-gx9uu]

I wonder about it too, how to make such a video?? I think it would be fun to use this kind of video in my class, anyone can explain? Or maybe give me the link to a youtube video about how to create this kind of video.

[@oscargr_]

Question: k is a coefficient indicating a restoring force depending on displacement x c is a coefficient indicating a resistive force depending on velocity, (workings against the displacement) The why in the final solution are both terms working in the same direction?

[@carultch]

They are not necessarily in the same direction, but in the special case that the mass is moving away from the equilibrium position, they are. If the position is x=2 and the velocity is x' = 1, then the restoring force is acting against the fact that it at a positive x-position, and the drag force acts against the fact that it has a positive x-velocity. The two forces are therefore both to the left, at this particular instant in time. So if x and x' both have the same sign, then the restoring force and drag force will have the same sign.

[@SRIMANTASANTRA]

Hi Professor Steve, tremendous and excellent lecture, Thank you

[@rcv3208]

U bar can be discontinuous function please explain Heaviside or Dirac Delta

[@黃祐宇]

how did u flimed the video, like are you writing on a glass? Then how did u fliped the writing, and the glass just has 0 light reflections. Pls tell me how I've been thinking of this question while watching many of your viedos.

[@federicogasparv]

Where is the next video? I would like to know how you transform PDE into ODE.

[@slogeshwarar4957]

the fact that laplace was the examiner of napolean bonaparte i knew it before steve brunton told it was amazing

[@mehmet4301]

Mr. Brunton hi, thank you first for the informative content. I have a question. You said that k is too big and so there will be no oscillations and the system will overdamped. But, for the damping of the system, do not we need to directly corelate to "c" ; k as stiffness and c as damping. Am I wrong?

[@yuma5297]

i am astonished that you usually dont learn that in Undergrad Physics. The Laplace Transform can really save much time.

[@Eigensteve]

Lots of undergrad physics curricula do cover this. But my experience is that it often really sinks in the second time students see it in grad school.

[@MrSandman213]

@@Eigensteve I believe I only first saw Laplace transforms when I took control theory at the University of Pennsylvania (which was a grad level class). Then it helped me in my math classes as an alternative way to solve differential equations. Never made any sense to me why the Laplace transforms weren't just taught earlier in math or physics curricula as opposed to learning them in a graduate level engineering class. They're super useful.

[@michaelbuckley4617]

@@MrSandman213 For what it's worth, I learned Laplace transforms as a sophomore undergraduate in Physics.

[@zyrtechy]

Is there something like this for systems of ODEs? e.g. two-variable models?

[@MrAri-gx9uu]

Wonderful explanation sir, thanks a lot

[@user-mf7li2eb1o]

Honestly i always wonder about his format. Is he really writing it that well or was there some post processing. If this was ot post processed he’s an actual genius (even more, he already is a genius 😂)

[@zhyfn9773]

what happens if k and c is also x-dependent or implicitly t-dependent?

[@Nikhil-uu9ov]

Steve these videos are awesome

[@jtlee1108]

Hey Steve, first off your videos are great! I am a fellow educational KZbin (nowhere near your scale), and I was thinking about making a mini series about some controls theory topics. Are you planning on continuing your Laplace topics? I am thinking a good place for me to start with the series I’m considering would be with how maths work in the Laplace domain and what it’s implications (for things like CLTF) are. If you are covering this, I will probably just skim over it and link to you- no need to reinvent the wheel. But if not I would love to do some episodes on that, but would need to start writing scripts in advance. Anyway, not sure if you will see this, but if you do It would be great if you would just let me know your thoughts. Thanks so much! And PS, my video style is super different (I animate everything and tend to do broader overviews aimed at getting a basic intuition, you do MUCH better at fine details), I would love to try and collaborate some time since they are complimentary styles if you would be interested

[@baris4321]

Mr. Brunton, could you please go on with a force function example to this video? Thank you for your efforts by the way...

[@carultch]

Here's how it would work with a forcing function. Given the free response equation of motion of: x" + 5*x' + 4*x = 0 x(0) = 2 x'(0) = -5 Suppose the force function (right hand side) were 10*cos(2*t), with the same initial conditions. x" + 5*x' + 4*x = 10*cos(2*t) Find the Laplace transform of each term: (s^2 + 5*s + 4)*X - 2*s + 5 - 5*2 = 10*s/(s^2 + 4) Shuffle initial conditions to the right: (s^2 + 5*s + 4)*X = 10*s/(s^2 + 4) + 2*s + 5 Rework the RHS, so we have one single fraction: 10*s/(s^2 + 4) + 2*s + 5 = (10*s + (2*s + 5)*(s^2 + 4))/(s^2 + 4) Expand numerator: (2*s^3 + 5*s^2 + 18*s + 20)/(s^2 + 4) Isolate X: X = (2*s^3 + 5*s^2 + 18*s + 20)/((s^2 + 4)*(s^2 + 5*s + 4)) Factor the quadratic: s^2 + 5*s + 4 = (s + 1)*(s + 4) Thus: X = (2*s^3 + 5*s^2 + 18*s + 20)/((s^2 + 4)*(s + 1)*(s + 4)) Set up partial fractions: X = A/(s + 1) + B/(s + 4) + (C*s + D)/(s^2 + 4) Heaviside coverup finds A and B: at s = -1, A = (2*(-1)^3 + 5*(-1)^2 + 18*-1 + 20)/(((-1)^2 + 4)*covered*(-1 + 4)) A = 1/3 at s = -4, B =(2*(-4)^3 + 5*(-4)^2 + 18*-4 + 20)/(((-4)^2 + 4)*(-4 + 1)*covered) B = 5/3 Construct result thus far: (2*s^3 + 5*s^2 + 18*s + 20)/((s^2 + 4)*(s + 1)*(s + 4)) = 1/3/(s + 1) + 5/3/(s + 4) + (C*s + D)/(s^2 + 4) Let s = 0, to solve for D: 20/((4)*(1)*(4)) = 1/3/(1) + 5/3/(4) + D/4 5/4 = 1/3 + 5/3/4 + D/4 5 = 4/3 + 5/3 + D D = 2 Let s =1 to solve for C: (2 + 5 + 18 + 20)/((1 + 4)*(1 + 1)*(1 + 4)) = 1/3/(1 + 1) + 5/3/(1 + 4) + (C + 2)/(1 + 4) 9/10 = 1/6 + 1/3 + (C + 2)/5 C = 0 Laplace result: X = 1/3/(s + 1) + 5/3/(s + 4) + 2/(s^2 + 4) Invert the transform to find solution for x(t): x(t) = 1/3*e^(-t) + 5/3*e^(-4*t) + sin(2*t)

[@Kuratius]

This madlad taught himself to write mirrored just to teach his students better

[@florianl7644]

He's not writing mirrored, it's just flipped

[@funwithstat]

The solution i think is : ce^-4t+ce^-t . Without the term c you can not find out the solution with the initial condition

[@Troll-cl1fh]

You are right, but c1=c2=1 because of initial cond-ns

[@funwithstat]

@@Troll-cl1fh thanks

[@sebastiansanfunas4674]

Thanks, great video!

[@MrSchubiduuu]

This is the capitalistic opponent of traditional Universities. And I know who is going to win this battle :-D

[@nguyendoanhoang1115]

Can anyone show me what does s stand for

[@carultch]

It's complicated. The s doesn't necessarily stand for anything in particular that starts with the letter s. You might think of it as standing for "state", I don't know if this is the actual term it stands for. What s represents, is complex frequency. The real part of a complex frequency is an exponential decay, while the imaginary part usually comes as a complex conjugate pair, that corresponds to oscillatory frequency of the sine and cosine terms. What a Laplace transform is, is a spectrum of sine waves, cosine waves, and exponential decays, that add up to the function in question.

[@fgrillo239]

If your shirt had writing on it that did not display mirrored text, math would no longer be the same for me

[@ancha20180]

thanks prof very helpull

[@valeesi]

It should be a crime to teach math as a shit prof. Why does every professor need to teach the same subject differently when there are such amazing contents available already? Let's crowdfund Steve and 3blue1brown 1 billion dollars to make lecture videos for all uni math classes on Earth for the next 100 years while most of our math professors could focus on supporting the students instead on answering questions, studying proofs and tutoring exercises. I almost never go to lectures because most of my math professors make me want to hang myself in the middle of the lecture hall, but I don't do it because great textbooks as well as amazing videos like this exist.

[@williamtell6750]

Excellent!

[@jimpollard9392]

Very nice.

[@007myzorro]

BRAVO !!

[@JohnSmith-lw2bm]

I think you have a cold. Great lecture though!

[@ifernandez08]

[@Eigensteve]

Thanks

[@AdoobII25]

first

[@amiaddvir8130]

I am an experienced engineer and my daughter is now going to attend one of CA leading Engineering school.. i was looking for "good stuff " for reference for her and myself, stumbled on your channel and lectures... Watched one .. then two ... So i completed the F.transform serie ... And found myself feeling sorry for not having seen such good lectures when i was a student .... Then I laughed as these set of lectures really show that in person lecture can not be so immersive. This medium of glass projection , the fact that you dont need to turn your back to the class , while writing, sharing slides, computer sims, also means that you dont need to limit yourself to pre defined set if slides. ++ You are very clear and able to take a topic that usually perceived as technical tool and breath life into it . Awsome!!

[@wibulord926]

took me many year and watching many time to understand from high school to 2 year college. But when i understand every thing become very clear lol