Laplace Transform: First Order Equation
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- Опубликовано: 5 окт 2024
- MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course: ocw.mit.edu/RES...
Instructor: Gilbert Strang
Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solution back to the ODE solution.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
"The purpose of a Laplace transform is to convert a differential equation into an algebraic equation." Well said, prof.
@123 321 Rarely stated by teachers
@123 321 not really. It’s basic knowledge that you sound like you belong on the subreddit /Iamverysmart though :)
That's true in ODEs, but not in PDEs.
It depends on your application, in electrical engineering we use it for signal analysis and also, it can be seen as an extended Fourier series . However this is only true when the double sided Laplace transform is taken into consideration, where s is a complex not real variable. The Laplace transform has many other applications aside for solving DEs. Another example is the property of its convolution equalling to multiplication in the S domain.
@@astroandriodrox2356 In Electrical Engineering the Laplace Transform is used for analysis of systems, while Fourier Transform is used for signal analysis.
Gilbert Strang ... what a fantastic educator. Hats off to you sir.
Nice picture if you look closely in mine you can see Nietzsche in there
Let me just say, that my mathematical skill have improved three-four fold since I've been working our out your linear algebra lessons, along with other mathematical concepts. Thank you Prof. Strang!!
Love you professor!
tienes redes sociales? soy de Perú tamb. tengo los mismos inetereses!
This is probably the best Laplace video and I am saying this after watching 5 or more videos on this same topic and not understanding a thing.
I wonder why, though. Dr. Strang didn't go into the intuition behind Laplace transform at all, which is what you need to have to even understand it in the first place.
@@AnindyaMahajan if you haven't already see "where the laplace transform comes from" parts 1 and 2 by Arthur Mattuck, MIT
ruclips.net/video/an5E940fqZQ/видео.html
Try this one. Herbert Gross is one of a kind as a distance lecturer.
Ass Möde Thank you!!,
@@Zack-xz1ph Thanks man
While the videos with colourful animations on the subject looks great, this the only way to learn. You never get to the depth of mathematics in those videos. Thanks Professor!!
With these videos I'm managing to truly see the connection between what I had seen in different courses at my university
I've studied advanced calculus for years but this is the best description and explanation so far.
Amazing guy, a treasure. I followed him more than 10 years ago and I still learn
Thanks MIT for sharing such wonderful Gems.
I love this why am I just seeing this…
"Do you realize what we've done?" I got chills.
I am so grateful to MIT and their excellent pedagogy for proving us with knowledge through MIT Open Course Ware.
Amazing Prof Strang! Indeed, a professor who knows how and what to teach. I plead to you for help. When I was in fourth grade we learned to extract a square and cube root of a number. It has been so long and one gets used to calculator, that I no longer know how to do that. No one in the RUclips community has posted. I am have a feeling you are the indicated person to refresh my memory. With all my respect, I bid you peace. Muchas gracias!!
Carlos Ivan Saa you learn such stuff in 4th grade?..
Very late but Tibees did a video on this!!
Carlos, take a look at Newton-Raphson iteration. It lets you approximate square and cubic roots - and much, much more!
I've watched several videos on this topic, but this one is the first one that explains in a few minutes what it's all about, what the purpose is, how to use it and how it works. I probably would have studied better at MIT, then I would have understood the Laplace transformation 30 years ago. Thank you!
That was an amazing amount of topic coverage for only 2 blackboards full. Great!
Finally after so many years I could understand the use of LT...thanks Prof for the enlightenment 🙏
Prof. Strang, thank you for your great teaching.
Gil Strang is one of the best educators that have ever existed on earth.
Brings me back to 1964 when I was taught Laplace Transforms by a wonderful teacher in India - as good as the MIT professor. I have fulfilled my long lost desire to learn from a MIT professor. I am tickling myself...
which teacher in India? I'm curious. IITB?
@@ashishkumarsharma1323 His face is flashing across my eyes.. but my memory is failing me terribly. It is a shame, of course, not to remember the names of your great teachers.,.. find me guilty of that ..
Our instructor spent an entire lecture on this. What's so wonderful is that he used the same materials, i mean examples, and I didn't understand a single word.
Superb clarity ,crystal clear.sincere devoted professor.
Oh God! This is a really cool explanation of the Laplace Transform . Thank You Dr. Strang
I can say that a high school student or a post graduate student would learn something from Professor Strang by just watching a video of his.
أتفق.
Profs in OCW videos almost always use really good chalks. They make such a nice sound, these chalks.
The Laplace Transform of e^(at)=1/(s-a) only true when s>a, so that means we can only conclude the y(t)=y(0)* e^(at) when s>a sir, what about s
If s
Thank you! I don't know why my professors made this so hard to understand. Yet Profs Strang lays it out nicely.
Because they suck at teaching?
I passed my course but I haven’t really understand it until I causally click this video. Thank you RUclips and professor.
Enthusiasm, passion. Something many teachers don't have!
very easy to understand. Such an outstanding lesson.
WOW !! THANK YOU Professor and GOD Bless you always.
This video is fantastic, easily the best L.T video I've seen
that was the best class i've ever taken, sir thank you so much
Sensacional....falou o necessário e simples👏👏👏👏em 4 minutos explicou a transformada de Laplace do jeito direto e simples, parabéns 👏👏👏
I really enjoy this brilliant man explaining advance mathematics. Just brilliant.
Yes, Prof Strang does a good job of explaining how to calculate Laplace transforms. However, no body every explains how or why decided to multiply f(t) by exp(-s*t) and integrate?
What was going through Laplace's mind? Why did he do this? What problem was he trying to solve? I understand that exp(-s*t) can be any frequency or decay.
I understand it as an extension to Fourier transform that adds decay over time to insure more functions' transforms converge
Laplace Transform is one of the best things ever
OH MY GOD, THIS IS GILBERT STRANG?! I LOVE HIS WORKS BUT I NEVER SAW HIM
Thanks professor.
Always inspired by prof. Strang. Really wish to meet you in person.
Just so freaking brilliant, that's how you teach
I knew it! It's clear from a quick comparison of the comments to the video that you are striking a good portion, probably a majority, of your comments. I wonder how many others have said the same thing.
Thank you so much for sharing!
How can you not love this guy?!
Your explanation is crystal clear thanks prof.
This is a solid Laplace transform video.
Very usefull, especially when you might forget a few details along the time...
Very well explained Sir thank you!
The best Laplace video I have seen. Thanks...
great blackboard sessions, thanks to Gil Strang's excellent teaching method
You made it 10 times easier for us, Sir. Thanks a million.
Sir you're too good ,thank you it was very useful
What a discussion Prof. Strang! Literally smiling the whole time. The discussion’s that good!
Thank you so much Prof. Strang
For learning the Methods of Mathematics I turn to Gilbert Strang.
You are one of the greatest mathematicians of era
I heard they are 2 of major founders of MATLAB
I would sell my soul for a chalkboard like that
LEGENDARY material
I am blessed , I could find this lecture 😇
Laplace, Lhopital, Coulomb, ........... Everyone must thank these and those French geniuses.
Who IS it. This french please merci d'avance cordialement FrèdéØ ČrèdéÔ CöœL bisous
I learnt more in 20 minutes than during 180min lecture
HE IS SO SWEET i wanna hug him
Inverse Lap Trans.:
Why:
Don't want the transform of the answer, I want the answer
How
We are in the s domain
Transform the answer to get the answer
(Get out of a back to t)
Particular:
What is t variable that makes y(s)
(Integrate
1/s - a
Numerator 1 constant
[So its form is Linear no problem]
Table look up to find the answer
Nice to see things haven't changed much since I learned this back in 1976. My prof had a Texas drawl and it came out as "poes and zerooos." it was then on to missile pitch stability analysis. That summer it was the bi-centennial. Girls were very patriotic back then. Got my flag poes raised in honor of the country.
this is so wonderful!!
GOAT - linear algebra, laplace, approximation etc etc etc
The professor's chalk board font size is a tad too small.
get out😂
that's what happened when they are trying to record a video and limit to only 1 (2) board(s)
3:19 "I will look only at S's that are bigger than A." Can someone please explain to me why this is justified?
@Tzabek So in other words we define s > a to make it work. It reminds me of the chicanery they indulge in in some aeronautical engineering texts I've read, where they integrate something and arbitrarily define the constant of integration as zero. Clearly it's valid, because the airplanes they designed using the math were historically known to work, but it always seemed a bit dishonest to me.
@@Ensign_Cthulhu It isnt that you define it to work. The transform takes a function of 't' and outputs a function of 's'. The domain of the new function of 's' is all 's' for which the integral exists. This is the case for all transforms.
Its kind of like when you take derivatives. The formulas are only valid for x values that the derivative actually exists. You just say derivative of ln(x) is 1/x. But the formula is only valid for x's that you actually have a derivative. In this case, x>0. Even though it is perfectly reasonable to plug -1 into 1/x after the fact, it is nonsensical in terms of the derivative.
You are amazing Professor !!! We are super GOLDEN :)
Thank you Sir for these amazing videos and would really appreciate if these lectures/videos were is some kind of order. Its difficult to follow..
ocw.mit.edu/resources/res-18-009-learn-differential-equations-up-close-with-gilbert-strang-and-cleve-moler-fall-2015/differential-equations-and-linear-algebra/
Welcome to university
Brilliant teaching, extrodinary
Where the mathematical equations in Dr Barnhardt's office in the movie: "The Day the Earth Stood Still" real? ....or were they gibberish?
Great explanation sir
Thank you so much professor
I have to go: thus, this not fully reviewed; however, this is also helpful with series notation, etc.📚
sir.....actually u didnt said y the minus sine used up there.....
at 3:18 why are we limiting ourselves to s>a ?? what about s smaller than a ?
Omar Al-Ani the limit of n-> infinity of e^(a-s)n is zero when the exponent is negative. a - s < 0 => a < s.
Why do we consider s greater than a?
This man is great ❤❤
I was with you for the first 5 seconds.......... I think.
He is calculating the Indian road traffic logistic.
what about for s greater than 0 in Laplace transform?
But what happens when a = c???
How can we know that ye^(-st) is zero in infinite? (At 8:58) What if y = e^(2st)?
We are putting the limit t=infinity.
e^infinity=0.
Sir which book you have preferred for this
I heard "we wanna find why and we know if" when he was actually saying "we wanna find y and we know f" xD
Dumb question... but why are we assuming s to be larger than a?
﴿اللَّهُ لا إِلهَ إِلّا هُوَ الحَيُّ القَيّومُ لا تَأخُذُهُ سِنَةٌ وَلا نَومٌ لَهُ ما فِي السَّماواتِ وَما فِي الأَرضِ مَن ذَا الَّذي يَشفَعُ عِندَهُ إِلّا بِإِذنِهِ يَعلَمُ ما بَينَ أَيديهِم وَما خَلفَهُم وَلا يُحيطونَ بِشَيءٍ مِن عِلمِهِ إِلّا بِما شاءَ وَسِعَ كُرسِيُّهُ السَّماواتِ وَالأَرضَ وَلا يَئودُهُ حِفظُهُما وَهُوَ العَلِيُّ العَظيمُ﴾ [البقرة: ٢٥٥]
(255) Allāh - there is no deity except Him, the Ever-Living,[98] the Self-Sustaining.[99] Neither drowsiness overtakes Him nor sleep. To Him belongs whatever is in the heavens and whatever is on the earth. Who is it that can intercede with Him except by His permission? He knows what is [presently] before them and what will be after them,[100] and they encompass not a thing of His knowledge except for what He wills. His Kursī[101] extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High,[102] the Most Great.[103]
[98]- Whose life is perfect, complete and eternal, without beginning or end, and through whom all created life originated and continues.
[99]- Dependent on none for His existence while being the sustainer and administrator of all created existence.
[100]- Allāh's knowledge encompasses every aspect of His creations in the past, present and future.
[101]- Chair or footstool. It is not to be confused with al-ʿArsh (the Throne) , which is infinitely higher and greater than al-Kursī.
[102]- Above all of His creations and superior to them in essence, rank and position.
[103]- Whose greatness is unlimited, beyond description or imagination.
- الترجمة الإنجليزية
very good lecture.
Doesn’t his example at 9:00 assume y(t) grows slower than e^(-s*t)? Please advise. Ps, great lecture overall 😊🙌🏽
I think its because e^(-st) is decaying, so it doesn’t grow at all?
In general when doing Laplace transform Re(s) (real part of s) must be large enough to ensure f(t)e^{-st} is decaying to zero as t grows
So Good!
Is there no duster?
amazing class
why the e^(-st) and no other?
Such a great video. Thank you so much!
thank you, sir
The explanation is clear and concise.
Thank you!
at 8:58 what if y fn is inf for t= inf ??
What did he do at 18:53. The answer is not same as 1÷[(s-c)(s-a)]
Hi, this is a partial fraction expansion(decomposition). He is good enough to do it in his head:), if you follow the rigorous procedure, you will get the same result as he did.
Given 1/((s - c)*(s - a))
To convert it to partial fractions, we form an equivalent expression in the form of:
A/(s - c) + B/(s - a)
To find A, cover up the s-c factor, and let s = c so this factor equals zero. Evaluate what remains, to find A. Similarly for B, except s = a:
A = 1/((c) - a) = 1/(c - a)
B = 1/(a - c) = 1/(a - c)
Thus we get:
1/(c - a) * (1/(s - c) - 1/(s - a))
Wish I saw this twenty years ago
nice and clear
But isnt this fourier transform
Here s is assumed a real variable. Fourier would be with an imaginary s (and integration from minus infinity to plus infinity).
The Fourier transform is a special case of the Laplace transform, long enough after the initial conditions, after all transient terms decay to zero, or close enough to it.
3:16 why S is bigger than a? 🧐
In order to be able to evaluate the integral, it must converge, implying that the coefficient in front of the variable t must be negative, i.e. s > a.