Intro to the Laplace Transform & Three Examples

[@mbenitez6722]

The wind is the soul leaving my body as i learn Laplace Transformations

[@samuelraj1186]

😂😂

[@ossahmadrezaazimikohnabi5108]

I was thinking the same thing 😂😂😂

[@apbianco]

The juxtaposition of the howls and the seriousness of the exposé is absolutely hilarious - you can't make that up. All in a sudden, I want to re-read Ginsberg poetry.

[@hisham_alhakimi]

هههههههههه

[@ronaldmadican2393]

It's the z transform next, and then you will have the joy of discrete signal processing! I envy you, I just loved that so much. Just think for a minute, you have all of these new vistas opening up for you to explore. If it pains you, then you are on the wrong course.

[@pres1dent1]

You can use a Fourier transform (special case of Laplace transform) to filter out the wind noise in the video.

[@sander_bouwhuis]

This deserves an award! LOL

[@user-jm6gp2qc8x]

It's a varying frequency, I don't think so

[@DrTrefor]

lmao:D I didn't think of this so I just got a new office and a new mic instead:D

[@pres1dent1]

@@DrTrefor Ha. Nonetheless, thanks for the great videos!

[@infinix2003]

@@DrTrefor lol, we should be able to practically implement what we have learnt

[@Warkip]

some say you can even hear the screams of the horrified students...

[@mohituniyal7]

I really heard some sound oooooooooooooooooooooohhhhhhhhhhhh

[@rapden18]

0:42. Bruh😂😂😂

[@LF58888]

Waahhhhhhgggg

[@tanmoyhaldar138]

Lol😂

[@johnmcintire3684]

Once it hit me - this prof looks and sounds just like my barber - the subject got a lot easier.

[@adeoladaniel]

Honestly.., been seeing commendable comments so far but as for me I rather feel ur not breaking this down enough and rather just jumping into solutions without even telling our it was brought about in the first place

[@Hr1s7i]

I was thinking the same thing, then I realised our man here is being very specific about the topic he is discussing. One is expected to already have mastered primitive functions and integration. If you look at it from that angle, it makes perfect sense that the format of the video is what it is. It would make for a several hours long mammoth of a video if he had to explain this by starting from the law of identity. Besides, you only need to look at it and you should be able to tell it's mechanics, if you've done any meaningful integration in the past. After that, all you need is to cobble together a few lines of code and never have to touch this ever again.

[@devalon8568]

You are one of the few that made a proper series of the Laplace transform. Much appreciated. Keep up the good work!

[@wunboonail]

The Wikipedia article on this topic freaked me out. It is so outstandingly presented and I like his style.

[@anonymouswolf4916]

The video becomes more exciting because he is happy to explain the topic.

[@brandonmohammed9092]

I was just starting my journey on laplace today and i love that you uploaded this today. Honestly on of the best yt channels there is. Keep doing great things sir because you make a great impact

[@brandonmohammed9092]

@@DrTrefor Thats great to hear, its sad honestly about this outbreak however it is really awe inspiring how all of us are coming together for this. Keep up the great work sir, you have helped me and many others and you will help more.

[@havory6621]

How did it go? Where are you now in terms of math?

[@WallyWhyte]

​@@brandonmohammed9092 I'd like to know too.

[@forrestgump1379]

Your enthusiasm makes your video much more interesting.

[@DrTrefor]

Glad to hear that!

[@stevanmiletic9780]

Shout out to Pierre-Simon Laplace for this life hack

[@carultch]

Laplace didn't come up with this method. He had a similar transform that is more like the modern Z-transform. It has properties in common with the Laplace transform, but isn't the form we know today. Heaviside and Gustav Doetsch are the ones who came up with what we call the Laplace transform today. The Fourier transform is rightly named in honor of Fourier.

[@theevilcottonball]

Yeah its actually lap lace transform. Back in the days women wore lace skirts and they bunched up in their lap. So they invented a transform for solving the mechanical PDE of the skirt folds, hence the lap lace transform was born.

[@CristianIonita-nm6xb]

@@theevilcottonball Excellent comment!! You made me giggle.

[@Defathomize]

How am I even supposed to understand something that's not fully explained (anywhere), like no one bothers to explain what even the purpose of laplace transforms is, you're just supposed to do it. Yet that's what I'm graded for and even if I get a good grade I would still have no clue what I'm actually doing. Kind of bizarre.

[@boushraadam6984]

i can relate to this so much lol

[@emersonbessler5686]

They are useful when predicting the performance of mechanical systems. ODE’s are needed to solve these types of problems and often involve initial conditions, and complicated systems have complicated ODE’s which require the Laplace method in order to effectively solve them because all the derivations get very complicated. That being said I have no idea how to solve these things.

[@mathiasensimon]

Its super useful for mathematically modeling systems for simulation. Eg control simulation in simulink

[@CautionRamen]

If you want a good understanding, you might like the Laplace transform video that Zach Star made

[@JR-iu8yl]

Cheers for these vids im currently doing Laplace Transforms for Maths Undergrad so this came at a perfect time.

[@daviddacosta1673]

What are you doing step function?!

[@paschikshehu7988]

It helps illustrate concepts since its values are 1 and 0 (it's also causal).

[@moayad80]

@@paschikshehu7988 bruh

[@sowickk]

it's helping you out since you're stuck

[@nol2521]

@@sowickk Hey step function, you must really... like math, huh....

[@BoZhaoengineering]

Laplace transform is very important when you try to design a dynamic system.

[@im_cpk]

But , what is Dynamic System?

[@orueom7720]

@@im_cpk a system you're designing or modelling in which parameters change over time. For instance, in chemical engineering you use laplace transforms to design reactors and model their reactions so you know how big the reactor should be, what the reaction conditions are etc.

[@BentHestad]

This is an excellent, little lecture. Thank you Sir, for this and other fine series in the field of mathematics!

[@DrTrefor]

You're very welcome!

[@enesozgry]

The hum in the background adds a vast loneliness atmosphere. I've got different emotions while listening this lecture and lost in deep thoughts.

[@gary1679]

you are a good man, thank you

[@SHAHHUSSAIN]

I just say ....outstanding❤❤

[@surendrabarsode8959]

It has been ages since i learnt and later forgot about this topic. I am now looking forward to re-learn it from you. Please speak slowly throughout so that it becomes easy to understand your words. Except for this, you are simply wonderful. Can you give examples of application of Laplace Transform in financial mathematics?

[@bhatusonawane7054]

Bro just play the video on 0:75x speed ....that's good to understand us.

[@nixonkutz3018]

Where was this video 40 years ago during my undergraduate diff eq class? I recall it being much harder, including the gamma function giving me cold chills down the back of my spine

[@soccerchannel9930]

your presentation is awesome

[@DRMath]

I remember solving these problems in undergrad!! Well explained Happy Teaching!! ✌️✌️✅

[@ZatoichiRCS]

Thank you for your effort on this video. You should start with the Fourier Transform. Even better is to start at the Taylor/McLauren Series. Can’t expect newbies to relate to this in depth material.

[@Zinxiee]

That comment halfway through about the howling wind made me laugh out loud. Thought it was just me going mad 😂😂

[@nanahaha-y2g]

I got 90 on my math systems exam. Hurrayy!!!

[@prithvikiranpremkumar9292]

Excellent video sir.

[@mufaafsal]

I was doing a video on this topic. I referred to this just for additional knowledge 😊

[@pragalbhawasthi1618]

Knew I'd love it before I even watched.

[@kianvaziri6939]

Outstanding presentation! Incredible clarity. I never knew that the gamma function is the connection to the factorial, thank you so much for making this!

[@j.o.5957]

Ah, yes, beginning yet another one of your series. Amen.

[@DrTrefor]

haha you are crushing these, did you make it all the way through vector calc?

[@j.o.5957]

@@DrTrefor Yes sir. I feel like I have most of the intuition down, now I just need to amass a large amount of solving problems. Probably work my way through a few previous exams, that should do the trick

[@collegemathematics6698]

Hi dr.Trevor , s is a complex number in general. And the complex numbers are not ordered set. Threrfore we can't say sa 4:03

[@carultch]

What he means more accurately, is that the real component of s has to be greater than a, for there to exist a Laplace transform of an exponential function, e^(a*t), in order for the improper integral to converge.

[@kimhughes1147]

Kudos Trefor - great contribution to subject - much appreciated

[@DrTrefor]

My pleasure!

[@allanolave2701]

All I can say is thank you very much, I love the way you explain.

[@shikha-qz8qi]

Very useful for me thanks u so much dear sir 🙏🙏🙏 Namaste because i am an Indian.❤❤❤

[@brycewalker1132]

Love your content and I am doing my dissertation on the theory and applications of Laplace, this is a great help!

[@DrTrefor]

Glad it was helpful!

[@mileslegend]

I like the explanation..will re listen this on repeat 🔁

[@visualgebra]

Professor, Your Affection with us greatful !

[@aadilashraf6592]

You Are The Best....I Can't Explain In Words...

[@Alannnn14]

your way to explain this topic is so good.

[@DrTrefor]

Thanks a lot 😊

[@Agnesshairsaloon]

You're the best Sir. The explanation is very clear, much appreciated

[@adrianspiby9969]

amazing explanation of the formula @ 2mins

[@hungryhippo420]

i was like, "okay interesting choice to play owl noises in the background of a math video" XD

[@marvelmayrandig1462]

That's really helpful and will be to everyone watching this pls continue posting vid like thse

[@hqppyfeet7513]

6:31 I don't understand where the "1" comes from. This is the part where I'm supposed to input "e^{-st} * f(t) dt" where f(t) = u(t-a), am I correct? How does f(t) become 1?

[@carultch]

The unit step function, u(t), is defined as an abrupt jump from 0 to 1, at the value of t=0. The general unit step function, u(t - a), has the abrupt jump happening at t=a.

[@MShazarul]

I read the comment and was wondering, what wind? And while going through the video, I laughed out loud! Haha good laugh!

[@karanbirsingh535]

Thank you for this great explanation!

[@AA-gl1dr]

Thank you for teaching!

[@crunchybanana6616]

my left ear really enjoying this

[@j.o.5957]

Hmm, interesting. Utilizing e^x's property to stay the same despite being integrated, such that you can integrate over and over again? Makes a lot of sense. Question to self: what other functions do that? The sine functions do something similar, which I guess allows us to display waves over and over again.

[@angelmendez-rivera351]

Sine functions are linear combinations of exponential functions, so no surprise there. If you have some polynomial of the derivative D, say p(D), and you have the equation p(D) = 0, then the solutions are going to be some linear combination of exponential functions. This is because the exponential functions are the eigenfunctions of the derivative operator.

[@mostafaahmadi4998]

Thank you SO much for creating this playlist. Would be greatly appreicated if you could kindly create a PDE playlist. Your videos provide an initution approach which are incredible.

[@DrTrefor]

I do plan to do more pde/Fourier stuff in the future:)

[@MDFarhanDEE]

Nice Explanation Thank you

[@DiegoAndrade]

MASTER CLASS!

[@Harry-ub2fv]

Please make a similar playlist on the Fourier series and Transform.

[@DrTrefor]

It's coming actually! About 3-4 months away. Finishing Vector Calculus first then moving to differential equations and it will be part of that playlist.

[@lishay4408]

At 7:15 how did negative s turn positive

[@markokuneye4530]

Because it is multiplied by a negative sign

[@emilycooper500]

Your cadence (the way you speak) is very helpful in retaining attention and making the material easier to stick with and follow. Thank you for the video!

[@user-wu8yq1rb9t]

Believe it or not, when I started to watch this video, I was in the bad mood. But now I'm smiling and my feeling is changing ... Great job ..... Thank you so much 💞

[@harishito]

that wind soul is scary as hell its 3 am i kept pausing if it was some owl omg

[@Thoalfeqargamer]

thank you for this amazing explanation. very well presented 😌.

[@DrTrefor]

Glad you enjoyed it!

[@aashsyed1277]

Danke you! Exellente explanation!

[@aashsyed1277]

not know whcih language this hehe

[@gateway5151]

Thank you for making this series. I was waiting for for from a long time. Thanks alot ❤

[@gateway5151]

It's a GREAT HELP. Thank you again.

[@mathhomeworkhelp1280]

Fantastic presentation! Outstanding explanation with excellent examples. 💯💯💯💯💯💯💯

[@DrTrefor]

Thank you so much!

[@dalisabe62]

What is the point of using the Laplace transform? Aren’t we supposed to transform back to what we started with? I thought the purpose of the transform is to make the integration of the initial problem doable.

[@DrTrefor]

Ya that's right. Basically you transform, then you do algebraic manipulations to clean stuff up, then you transform back. We do this a bit more further down the playlist.

[@DarkBoo007]

I am so damn mad that no one ever explained the Gamma Function and n! like that! I had to learn that on my own when I was in college (My Calc II professor was horrible). It was a good thing I did because when I took Differential Equations (Last semester in college), I had this insight and things were not confusing for me. I appreciate that you explained the Gamma Function with rich substance because many students do not get the explanation to why it is equal to the factorial.

[@erikawimmer7908]

Great explenation but i have got a question: if s is a variabel how can we then integrate with redpect to x? You can't integreat a function with two variabals with tespect to only one of them.

[@DrTrefor]

You can! Basically what you do is hold s as a constant and integrate with respect to x where you treat anything with s identically to how you would if it was a constant.

[@erikawimmer7908]

Thanks! I thought this was only possible with partial derivatives.(btw. sorry for the bad spelling I am from germany and I am only 14.

[@erikawimmer7908]

@@DrTrefor sorry that i have to ask u again but if we can treat s as a constant when integrating with respect to t coudn't we solve any differential equation like that (at least 1st order odes) . What i mean is coudn't we just multiply both sides by dx and then integrate the one side with respect to x and the other with respect to y even if the x and y terms are not sapetated?

[@WallyWhyte]

​@@erikawimmer7908hallo. Wie gut sind sie in Maths? Und was studiert sie?

[@lukeauslender6494]

@5:04 Would s=a make it undefined?

[@hqppyfeet7513]

Yes, well you would see that the result "diverges", which means that the limit either (1) does not exist or (2) reaches infinity.

[@malihabintehasan7182]

your videos helped me a lot! thank you so much

[@haidarasifi6169]

I am having problem finding a book to study the Laplace transform. Some people told me to use frank ayres's book but I couldn't find anything useful or anything similar to formulas in this video. Can you tell me what book I can read to have access to a lot of questions and problems for practicing?

[@ThePaperCreater]

Why does this video has 85 dislikes? It's so helpful

[@nathangmail-user8860]

ah yes, back in the day when we could all see the number of dislikes

[@ThePaperCreater]

@@nathangmail-user8860 There's an extension which has all the historic dislikes from before December 2021 and any new dislikes after are estimated from the current users with the extension, I'd recommend it 👍

[@andrewharrison8436]

Well, it gives people an opportunity to engage in the discussion and that in turn enables the algorithm to realise what a great video this is. Otherwise you have to wonder at people even clicking on a maths video when they obviously don't like maths.

[@jluke6861]

Great Video. Thank you.

[@keldhansen4071]

Thanks for the video. I need to understand how an exponent can be complex, s = σ + jω, and what it means. This is not explained. Also, as far as I know, Laplace transform is used to cenvert a continuous function in the time domain, into a function in the frequency domain. Normally, poles and zeros are presented in the complex s plane.

[@carultch]

To understand what it means for an exponent to be complex, it all comes down to Euler's formula, to make sense of the imaginary part of the exponent. Essentially, it rotates the number in the complex plane, instead of scales it, like a real exponent does. Given a general complex exponent of a+b*i on Euler's number, we can split the exponent with properties of exponents. a and b are real, and combine as discussed to form a complex number. e^(a + b*i) = e^a * e^(b*i) e^a is a positive real number, so it's just a scaling factor. e^(b*i) is what we unpack with Euler's formula, which gives us cos(b) + i*sin(b) What's behind Euler's formula, is the Taylor series. Use the Taylor series of e^x, and plug in an imaginary value for i*theta for x. We can do this with first principles of complex numbers, because a Taylor series is just arithmetic and integer powers. You'll get an infinite series of real terms with even exponents, and an infinite series of imaginary terms with odd exponents. These two series, are Taylor series of cosine and sine respectively.

[@aayushpatel8913]

Love from INDIA ❤️❤️❤️

[@manishjain1768]

The negative sign ( e raised to negative st) in the formula for laplace transform means exponential decay right? If not why else is e particularly raised to a negative power ?

[@carultch]

Yes, it does mean exponential decay. What the Laplace Transform is doing, is calculating a spectrum of complex frequencies, that combine to form the original function. The real part of these complex frequencies, is exponential decay. The imaginary part is oscillatory frequency in radians per time unit.

[@aashsyed1277]

fun fact: gamma of a integer is that integer factorial-1 ! that's how people define (1/2)! even that recursion is true for non integers how cool

[@mastershooter64]

using the same gamma function you can even do it for complex numbers!

[@upendownlinker]

mind_blown.png

[@danielduge3140]

This was amazing

[@EzzedineAli2ndSH]

You're acually goated. Thnx alot

[@ghasemmanouchrhti10]

clear explanation, thanks

[@kenny44871]

Great explanation, this all makes so much more sense now.

[@droomahbroo9291]

My two year old brain is loving 2:30 with the English(auto-generated) Subtitles on

[@domigo_8678]

I want to live this man's life. Doing maths on a a cottage by the jungle, where mythical creatures howls in distance!

[@akiiiphysics3345]

I'm in 10th grade like it... India

[@danielserrafreese4543]

What is bigger, n! or infinite?🤓 Thanks for this great video.

[@takey0208]

Thank you for this!!!

[@DrTrefor]

You're so welcome!

[@ethanhunt3419]

these Videos are so great helping me for masters# student of University of Windsor ontario

[@zeyadalsheikh3839]

With all do respect.. you had to focus on just Laplace transform and stick to it giving more examples about it. The Gama transform is another subject that confused me much while I am trying to understand Laplace, also the u function is confusing. Anyway.. your explanation is great. The winds give more horrifying feeling of the complex stuff. You could record your voice separatly and add it later to the video.

[@ravenarc3652]

"Whaa.... whaat are you doing, Step Function!"

[@محمدالشهري-ظ2ك]

Q. When you convert the DE to an algebraic equation why you have -2s+3?

[@devalon8568]

You get this after simplifying after plugging in the initial conditions.

[@mimomira8808]

love the explanation. what a cute and happy teacher

[@noahie1438]

i didn't even notice the wind noises until you pointed it out

[@johnbatchler8551]

Great job

[@bhoopendragupta4782]

Great video, easy explanation ❤

[@DrTrefor]

Glad you think so!

[@willsayswords3451]

great video 👍

[@jflopezfernandez]

Awesome video, thank you

[@brunobautista6316]

Is good, managed to easily understand everything. But, with all due respect, it lacks a lot in terms of explaining. I mean, is entirely theory, but nothing about how it comes to appear this Laplace Transform. I think you may agree with me that, when it comes to maths, there is ever a logic and somewhat simple explanation to the very reason because "a thing" is "created" (or, well, defined. You get the point). Integrals has all that Riemman's Sum behind, Taylor Series all that convergence thing behind, and so on so on so on, what I am trying to say is that there is a reason for "something to be like it is", and for newcomers or just people that doesn't fully understand this, the explanation (that it is almost always an "intuitive" explanation) could be of very very great help. Don't misunderstand me, the video is of course excellent

[@abelfernandes6862]

God Bless You Great Video . BUT Where Did The N Come From

[@knowledge90s93]

The laplace transform can be applied to both linear and non linear differential equation? true or false?

[@DrTrefor]

True!

[@realislamicguidance2375]

Laplace Transform converges (gives finite value) in ROC. How is this information (the finite value of LT) help us anyhow?

[@kalyanroy4180]

Sir, can you put a video for Gamma of half integers input and how really this gamma function was brought into this form .... you really explain very well

[@sander_bouwhuis]

When would I use a Laplace transform? Is it for when you cannot (easily) use 'normal' integration?

[@carultch]

Chances are, if you can't use normal methods of integration, you probably can't take the Laplace transform in the first place. It's value comes from differential equations, and particularly differential equations involving discontinuous functions like the unit step and unit impulse. It's common that you get a diffEQ in the form of y"(t) + b*y(t) + k*y(t) = f(t), where y(t) is the function we are solving for, b and k are constants, and f(t) is a given function of t. You can think of it like a mass on a spring with damping friction, being driven to oscillate by a forcing function f(t). When f(t) is a function like sine or cosine, earlier methods of differential equation solving work, like the method of undetermined coefficients and the second order homogeneous solution via the prototype exponential. But when f(t) is an exotic function like a piecewise function with unit steps, the Laplace transform has a great advantage. An application where you see this, is control systems engineering.

[@leandroevangelista4660]

What software do you use to do this equations's animation ? Thanks

[@DrTrefor]

It’s all just powerpoint;)

[@sandeepsai9437]

how you solved the differential equation that you showed first

[@carultch]

Given: y"(t) - y'(t) - 6*y(t) = 0 y(0) = 2, y'(0) = -1 Note that this can also be solved with the Ansatz method, assuming y=e^(r*t), and solving for both values of r. But to work with the topic, I'll show the Laplace transform method. I'll use the British pound sign to indicate the Laplace transform symbol. Assign Y(s) = £{y(t)}. Take the Laplace of each term: £{y'(t)} = s*Y(s) - y(0) £{y"(t)} = s^2*Y(s) - s*y(0) - y'(0) £{0} = 0 Assemble equation: s^2*Y(s) - 2*s + 1 - [s*Y(s) - 2] - 6*Y(s) = 0 Expand brackets, and shuffle initial conditions to the right: s^2*Y(s) - s*Y(s) - 6*Y(s) = 2*s - 3 Factor the left, and isolate Y(s): (s^2 - s - 6)*Y(s) = (2*s - 3) (s - 3)*(s + 2)*Y(s) = (2*s - 3) Y(s) = (2*s - 3)/[(s - 3)*(s + 2)] Setup partial fractions: Y(s) = A/(s - 3) + B/(s + 2) Heaviside coverup: at s = +3, A = (2*3 - 3)/(3+2) = 3/5 at s = -2, A = (2*(-2) - 3)/(-2 - 3) = 7/5 Y(s) = 3/5/(s - 3) + 7/5/(s + 2) Invert the transform, and we have our results: y(t) = 3/5*e^(3*t) + 7/5*e^(-2*t)