I wish people use Laplace more to solve second order linear differential equations as the initial values are easier to plug in. The advantage of converting differential equations to algebra makes it easier to deal with. Are you planning to do any Z-transforms as this may be interesting?

Hey Dr. Peyam, sweet video! I actually wanted to ask if you were ever going to make an in depth video on transforms? I feel like we take transforms for granted and wanted to see more of them. Love the videos and thanks!

Yes! Last time you talked about Laplace I went like, wait this is illegal I don't see any odes anywhere in this and I immediately thought about using Fourier transform in Laplaces place instead 🥺 Laplace just seemed so freaking magical when we studied odes but then abruptly disappeared forever, not even making a comeback when we got to pdes 😭

@drpeyam

2 жыл бұрын

I never use it, tbh 😂 But they’re still really cool

well i studied this about 40 years ago, and never had to use it in my professional life, but it's still there and it's amazingly fun to watch your videos :)

Ach! Professor Peyam, you are the best professor I ever met online.

Oh man this is SO 😎COOL👍 Dr. Peyam !! And curious for more about the Laplace transform turns and into more generally than the Fourier transform. .. and thank you for naming the techniques that you finger snap over for us less brilliant to explore on our own.

We can use laplace transform for analysis of comum eletrical circuits (wich involves capacitors, inductors and resistors as components). Using the laplace transform we can get the output of a system in the time (t) domain and using the transfer function in the laplace domain (s) we can get the output response of a system in the frequency.

Loved this one, it seems alot of fun doing the algebra part too!

this is a really cool way to approach this problem. solving this normally by just forming a 2nd order de and then solving can be long sometimes

thank you for the nice explanation. Helped me indeed.

Beautiful video!!! 😄😄😄

Cool way to evaluate Diracs equation for electron dynamics avoiding decoupling by further differentiation instead using matrices / Cramers rule

So cool. 😎

Nice video

Does the determinant you calculate for the denominator in Cramers rule have anything to do with the characteristic polynomial you would generate when normally solving for the Eigenvalues in this system? I haven’t checked it, but it felt eerily like an Eigenvalue calculation

@drpeyam

2 жыл бұрын

Yes exactly!

@DarkMonolth

2 жыл бұрын

Awesome! Interesting that even in this method the “standard” method comes out a little. Does this generalize at all to systems of 3, 4,…, n differential equations?

@drpeyam

2 жыл бұрын

Of course!!

Easy parcheesi. What is the computational complexity of this method for larger systems, compared to the usual methods?

what are the Hilbert transform and the z transform?

Thanks for interesting topic. We usually find eigenvalues and eigenvectors of the coefficient matrix. x'( t ) = 7x( t ) - 3y( t )　…① y'( t ) = 10x( t ) - 4y( t )　…② 5・① - 3・② implies that 5x'( t ) - 3y'( t ) = ( 35 - 30 )x( t ) + ( - 15 + 12 )y( t ) = 5x( t ) - 3y( t ). Solving this equation we have 5x( t ) - 3y( t ) = ( 5x( 0 ) - 3y( 0 ) )・e^t.　…③ 2・① - ② implies that 2x'( t ) - y'( t ) = ( 14 - 10 )x( t ) + ( - 6 + 4 )y( t ) = 4x( t ) - 2y( t ) = 2( 2x( t ) - y( t ) ). Solving this equation we have 2x( t ) - y( t ) = ( 2x( 0 ) - y( 0 ) )・e^( 2t ).　…④ By ③・④ we have x( t ) = 3( 2x( 0 ) - y( 0 ) )・e^( 2t ) - ( 5x( 0 ) - 3y( 0 ) )・e^t, y( t ) = 5( 2x( 0 ) - y( 0 ) )・e^( 2t ) - 2( 5x( 0 ) - 3y( 0 ) )・e^t.

@drpeyam

2 жыл бұрын

Right, I mention eigenvalues at the end

great video. question: is there be a video on solving pde using fourier transformation in the planning as it is often an elegant way to deal with the schroedinger equation

@drpeyam

2 жыл бұрын

No way, Fourier transform is awful for nonlinear PDE

thanks for the awesome video

@drpeyam

2 жыл бұрын

You’re welcome 😄

@SuperYoonHo

2 жыл бұрын

@@drpeyam 👍

Wow, I never knew how useful the laplace function was!!!😆😮

@drpeyam

2 жыл бұрын

Right???

Can we solve this by e to the power of a matrix method too?

@drpeyam

2 жыл бұрын

Yes of course, but this is more direct

Thank you for an informative talk. Whilst I think most of us will be satisfied that Laplace transforms DO work very nicely in solving differential equations, are there any references that might be useful to explain WHY they work?

@drpeyam

2 жыл бұрын

My lecture notes :)

@drpeyam

2 жыл бұрын

people.tamu.edu/~tabrizianpeyam/Math%20308/math308.html

@vaughanwilliamson173

2 жыл бұрын

​@@drpeyam Thank you for your reply and I had a bit of a look at your notes which I think should satisfy the intended audience. Still something is on my mind, - perhaps I wasn't explicit enough in my earlier comment. Likely many have the same question which is "where did the Laplace Transform (LT) come from?" I have seen a two part youtube where Arthur Mattuck of MIT explains the development of the LT from the sum of a power series. He comments that the LT is the continuous analog of the discrete valued summation. All good there... but still... doubts. I found that a paper by Deakin exists in two parts detailing the history of the LT. Sounds like what we need - except the premium to obtain the papers is probably not far off the price of a paper textbook! Eek!! Other sources like Wikipedia, Quora etc refer to Euler's contribution and Laplace's further developments, then Heaviside, but still not quite what I am after (and likely others as well). Very likely, what I am after - perhaps not really knowing exactly what it is I am seeking (!) - is an understanding of what insights occurred such that a) transforms of functions could be made, b) interpreting the properties of those transforms and c) how that insight led to the realization that LTs in particular could take differentials and integrals and manipulate them algebraically. Perhaps the insights into the development of the LT may be the WHY of my earlier comment. So do you perhaps know of readily available mathematical history references that might be appropriate?

From where we got (s-1)(s-2)

Challenge for you solve it in symmetrical form You solve system in symmetrical form when you solve first order PDE by method of characteristics

Is it possible to differentiate both equations, allowing you to cancel down to second order linear ODE? This seems to give the same answer

@lego312

2 жыл бұрын

rather, after differentiation, you can use the derivative to cancel the original eqns.

@xinpingdonohoe3978

2 жыл бұрын

If you differentiate the first function, then use the second equation to replace y'(t), then use the first equation to replace y(t), you have a second order linear ordinary differential equation in terms of x(t) and its derivatives. From here, you solve normally and use the first equation to get y(t). That's the way I was originally taught to tackle coupled linear ordinary differential equations.

here i am again cute professor........ looks like you gonna be leading me through my second semester exams

the only helpful video on youtube but i wonder why this method didn't work for my worksheet

3:48 how did we get this?

@drpeyam

2 жыл бұрын

It’s called Cramer’s rule and there’s a video on that :)

I have no Idea what this is XD

What is your book reference

@drpeyam

4 ай бұрын

I came up with it on my own but maybe consult Boyce diprima?

The Laplace of t^2 + 3, Times Laplace of the sine of t, Minus one over s, And invert the whole mess, Is t^2 minus cosine, you'll see

"Beautiful L" :)

Isn’t Eigenvalue method a lot easier?

@drpeyam

4 ай бұрын

This is more direct

"Beautiful L" 😂

That is waaaay too much work for a linear system

@drpeyam

2 жыл бұрын

Not really?

Your beautiful L looks a lot like a fancy Z

i cannot take u seriously my frend

@drpeyam

Жыл бұрын

Ok