Using Laplace Transforms to solve Differential Equations ***full example***

[@lythd]

my exam is in a few hours and you are a life saver!!!! thank you!!

[@MidwestSirenProductions]

You just explained how to do this ten times better than my college professor did earlier today. Thank you for the help!

[@balls4924]

bold of you to get clarification early instead of cramming before a test

[@nerd2544]

@@balls4924 sup brah, my final is tomorrow 💀

[@nerd2544]

ight i think im getting a 75-80. was way easier than previous years but still fucked up some questions

[@xspected5076]

@@nerd2544 Any update?

[@aznmayo]

@@nerd2544 bro did u pass, my grade is so fucking low i nee dto as man but I believe rahh

[@zlatanbrekke6538]

Easier way to solve partial fraction: just decide S to be the roots, for example S - 1 = A(S + 1) + B(S - 2) Choose S = -1 -2 = -3B -> B = 2/3 Choose S = 2 1 = 3A -> A = 1/3 Way quicker than solving a linear system of equations

[@10harinims61]

both are simple ways ... it depends upon us to choose which way to use

[@zlatanbrekke6538]

@@10harinims61 I guess it depends on what you are used to yeah

[@masonmccullough5242]

@@10harinims61 but one is simpler, you can choose the harder way if you want lol

[@TrinidaddyGdom]

Sometimes this method doesn't work where a system of linear equations will always work. But I agree, idk why anyone would choose the hard way lol My DiffEQ professor always tell us to be as lazy as we possibly can lol

[@wsar7669]

Thank you so much I got so stuck because I didn't understand his method at all

[@jamaljaffer8412]

This is one of the best maths videos ever watches, many thanks.

[@DrTrefor]

Glad it was helpful!

[@cuie6967]

I am excited after watching this, for no particular reason. Maths just amaze me:) Thank you for this high-quality video series! ( they are so well explained that even a high school student like me can understand!)

[@spyrosmanolidis8516]

Thanks

[@DrTrefor]

Thanks so much!!

[@xhydrous]

i dont even know why i show up to class anymore. i learn so much more out of these online videos than i ever will from class

[@Mockedarchie]

This was magnitudes easier to understand then the way my professor showed it. Thank you

[@brandonmohammed9092]

Wow, it's like if you're kinda doing exact equations, that's cool, gotta learn this more, thank you again so much for this!

[@allblueandyou]

really concise and clear explanation! thanks a lot buddy! 💙💙💙

[@ezraitejamile]

I am so grateful I found your channel tata 😭 God bless you!

[@SB-wk7cr]

Done well, really helped me put everything together during these covid self-teaching times.

[@TALCOHOME]

You are the best math teacher ever💥!!!.

[@cernejr]

Not bad, but I would like to see the explanation of what is going on under the covers. What was Laplace's thinking when he invented this transform? Same question applies to other integral transforms.

[@supremeleader5516]

If you found your answer them pls refer me source too! I seriously want to know

[@uhmody5796]

the whole point of the Laplace Transform is to make solving differential equations easier. going from transforming the equation from time domain to s domain, solving, and using inverse laplace back to the time domain.

[@CKNGAI-r8x]

The story I've heard is well to simplify it down. Laplace looked at the fourior transform and thought hmm what if I just made them converge and well it still works. So, he poblished it as his transformation.

[@kenodinson8323]

This is something I’m curious about just learning about them this week and am curious what the intuition is behind them

[@safeguard.mentality]

Advice to all math (and physics) students: don't go into any math (and physics) courses looking for conceptual explanations and intuition. You will be greatly disappointed. "Maths" and "Physics" has been technically run by "Derivators" (people who manipulate equations) since the 1700s, so don't go into university courses (or even a research career later down the road) looking for concepts and intuition. Carve out the time (and necessary space in your head) to devote to intuition and philosophical insight separate form mechanically performing derivations for your classes, and find like minded people to discuss and build your intuition with in philosophy of maths/physics circles and from expository popular maths/science books and online resources. Don't give up! The world desperately needs people who actually understand what phenomena is occurring and can communicate that to the general public, rather than speaking in jargon and insisting on notation to hide that fact, for a genuinely more scientifically literate society. I guarantee you, most Mathematicans and Physicists lecturers and researchers have very little intuition for the majority of topics they covered to get to where they are, and understandably so. We need more people like you to share your understanding with the world! So study hard in class and learn even harder outside of it!

[@OceanageMangoma-w4p]

Watched all your series videos ,they helped a lot thanks so much

[@camillejones9461]

Thank you! Your videos are so helpful while I'm taking DE online!

[@suponjubobu5536]

That clarifies a lot! I might not fail now!

[@killthem9669]

Yeah, I hope I will not fail tomorrow

[@marco.9900]

Did you pass?

[@suponjubobu5536]

@@marco.9900 Yes, and I graduated with first class honours earlier this year :D

[@connoratkinson8897]

Really saving my engineering ass before my midterm thank you :)

[@aznmayo]

if anyones confused on 7:50 , you can ignore everything he says and just do the laplace transform of Y(s)

[@daboyz6106]

Very well made and clear. Thank you.

[@johnewald6371]

Fantastic explanation!

[@clarinetowl7554]

Trefor, thanks to you I might justg pass my diffeq class. JK, I'm DEFINITELY gonna pass and it's cause of your diffeq videos. Thank you, and lots of love from happy valley.

[@tekfreak6128]

Great video and clear explanation thank you. I must say I find your approach to partial fractions my preferred method. However I do see approaches where the roots of the denominator are set to zero: s=-2 and s=-1 to get A and B. This raises the problem of division by zero. So how can this be mathematically sound yet gives “the right answer”?

[@continnum_radhe-radhe]

Thank you sir 🔥

[@vedantpratik9352]

Thank You Sir , Very Much Helpful Video.

[@cocothetimeless8382]

dude be saving math students azzes

[@ac-jk9mz]

this is awesome sir, thank you

[@jasonnatanaeldrummer]

Thank you so much sir!

[@crazygur1y]

ilysm

[@kakunguchitakwa4615]

Thank you this video really helped me !

[@wryanihad]

You are so polite i wish having happienes in your live

[@suhailawm]

sir post some limit sequense . converge or not. example videos

[@soumapriyamondal3808]

Here we assumed Y(s)=L{y(t)} and then at then did L^-1{Y(s)}=L^-1{L[y(t)]} to do the inverse... Will it work everywhere?? I mean can we apply it in every problem...

[@jennyskrytenjohnsen8776]

Great video! How to du know that L{y"}= s^2y(s)-sy(0)-y´(0)? Is there any intuitiv way to see this?

[@DrTrefor]

I walk through this in an earlier video in the Laplace playlist:D

[@matthiastakele]

Woah where can I get that t-shirt!

[@riss8858]

its in his amazon affiliate shop! a little different but still cool

[@debajitroul7239]

Love you sirrrr

[@aashsyed1277]

super new video wow!

[@MossesRoss]

Liked 🙂

[@aldoestekkerzz3705]

once you do the inverse laplace, dont you require a Heavside function?

[@حلفانكوارقينوكي]

Where has Laplace transform originally come from? How was it derived? A question that intrigues many!!!

[@SSNewberry]

Where did you get the initial t-shirt with the first and second derivatives on it?

[@jordanbrowne8481]

God bless your soul.

[@bitte929]

can someone tell me whats the use of the algebraic equation? is it just helping to go to the time domain or does it also convey some information and is our main goal of this laplace is to solve ODE and go to time domain?

[@safeegull22]

Here i have confusion, how it is 2 b, as we see put -1 as s so it will b -3b

[@nabusobahassan902]

Nice

[@suhailawm]

tnx alot sir.

[@nick45be]

In which case of differential equation I can't apply the Laplace transform? Or can I apply Laplace everytime I want?

[@carultch]

It is a valid step to apply Laplace transform any time you want, to solve differential equations, as long as you are in the domain where t >= 0. There is a bilateral Laplace transform that covers the general case where t is any real number, and many standard Laplace transforms also work for the bilateral Laplace transform, by coincidence. Whether or not it will help you, is another matter entirely. Some functions like secant and tangent, are not of exponential order, and have no valid Laplace transform, not even as an infinite series. In other cases, it may not be possible to reduce your result to standard Laplace transforms, in order to invert it. I've tried to find an example of a diffEQ that could be solved with L{ln(t)}, which does exist, but I've yet to find one that works. It works best for polynomials of t, exponentials, sines, cosines, Dirac impulses, Heaviside step functions, linear and/or multiplicative combinations of the above, and convolutions of the above. While it exists in theory for fractional powers of t and reciprocals of powers of t, it is much more difficult to use it in practice for solving diffEQ's.

[@MeysamHoseini-pj9en]

Hi, What is Laplace of ; k power x+iy , k is a Real constant number ? Thanks,

[@GuiTianao]

Dr. Bazett, where can i get the shirt? It looks so cool!

[@ammarhasnain7148]

How to convert integral to differential by Laplace

[@di-riso]

You could also just plug s =-lnx in

[@jerichokhaliq2648]

where can i get the t shirt your wearing in the start

[@despicableme7081]

Where I can get the proof of the Laplace transform of 2nd order derivative ???

[@DrTrefor]

Just apply the rule for first derivatives twice in a row

[@MinecraftStonewideos]

I love you bro

[@aayushmohan514]

00:00 nice shirt

[@abhishekvanenooru2869]

Shirt is kool where can I get it

[@j.o.5957]

Damn, this's hard. What level of math is this recommended for?

[@mathadventuress]

Differential equations

[@10harinims61]

it isnt hard ... dont give up ... keep trying... try to get the basic concepts ... u will definitely find maths easy

[@Jeff-xy7fv]

@@mathadventuress Yep! Diff-EQ is diff-e-cult!

[@SuperDeadparrot]

Can a Laplace Transform be used in a boundary value problem?

[@carultch]

Yes. You just have to be creative. As an example, suppose we are given y(pi/6) = 3 and y'(pi/4) = 1, to solve the diffEQ of y" + 4*y = 0. Let u = y(0), and let v = y'(0). Thus: L{y"} = s^2*Y - u*s - v And our diffEQ's transform is: s^2*Y - u*s - v + 4*Y = 0 Shuffle initial conditions to the right, factor the left: (s^2 + 4)*Y = u*s + v Solve for Y: Y = u*s/(s^2 + 4)+ v/(s^2 + 4) Multiply 2nd term by 2/2, so we have L{sin(2*t)} available to us: Y = u*s/(s^2 + 4)+ 1/2*v*2/(s^2 + 4) Take the inverse Laplace: y(t) = u*cos(2*t) + 1/2*v*sin(t) Now we have the general solution for any initial conditions. But we were given conditions elsewhere than t=0, so we now need to apply them, and solve for u & v: y(pi/6) = 3 = u*cos(2*pi/6) + 1/2*v*sin(2*pi/6) = u/2 + sqrt(3)/4*v y'(t) = -2*u*sin(2*t) + v*cos(2*t) y'(pi/4) = 1 = -2*u*sin(2*pi/4) + v*cos(2*pi/4) y'(pi/4) = 1 = -2*u Thus: u = -1/2 & v = 13/sqrt(3) Solution: y(t) = -1/2*cos(2*t) + 13*sqrt(3)/6*sin(2*t)

[@carultch]

Another way to be creative to use it for non-initial conditions, if you are given both conditions at the same point in time, is to use a change-of-variables to t-shift the problem, and then undo the shift.

[@Yamazakura00]

My mind is blown...

[@migueltrinidad736]

Is that shirt still for sale?

[@СнежныйБарс-г2я]

798//6.10.21

[@austinfritzke9305]

8:08 that equivalency statement doesn't provide any insight

[@10harinims61]

inverse laplace of transform of F(s) is f(t) right

[@10harinims61]

the same way laplace inverse of Y(s) is y(t)

[@iindombotrophy2777]

👍

[@ZeeshanKhan-xi8qt]

i am gay

[@BGHlovesmath]

need that tshirt

[@AODCRIB]

dfkm!

[@kennobags6904]

Heh as awaken my 3rd eye

[@princefresh7588]

why are u happy ... im mad bcoz of that im offended

[@triggeredsydney]

I would solve that differential equation instead.

[@BGHlovesmath]

laplace makes solving equations with a higher order easier

[@panashetigere3418]

Hatife takapinda muma lectures nemi mdhara ma tutorial enyu ma1 🙏😀🤌

[@crimsonred7517]

Thanks