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?
@factsheet49302 жыл бұрын
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 😭
@drpeyam2 жыл бұрын
I never use it, tbh 😂 But they’re still really cool
@mujimuji60022 жыл бұрын
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!
@pheldge88062 жыл бұрын
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 :)
@arturb29962 жыл бұрын
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.
@MrFlaviojosefus2 жыл бұрын
Ach! Professor Peyam, you are the best professor I ever met online.
@marccox89772 жыл бұрын
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.
@user-dl8nk5bf8v2 жыл бұрын
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.
@drpeyam2 жыл бұрын
Right, I mention eigenvalues at the end
@DarkMonolth2 жыл бұрын
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
@drpeyam2 жыл бұрын
Yes exactly!
@DarkMonolth2 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?
@drpeyam2 жыл бұрын
Of course!!
@NittyLittyNiturzion2 жыл бұрын
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
@brendanlawlor22142 жыл бұрын
Cool way to evaluate Diracs equation for electron dynamics avoiding decoupling by further differentiation instead using matrices / Cramers rule
@BigAsciiHappyStar2 жыл бұрын
Beautiful video!!! 😄😄😄
@vaughanwilliamson1732 жыл бұрын
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 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?
@CornishMiner2 жыл бұрын
So cool. 😎
@MushookieMan2 жыл бұрын
Easy parcheesi. What is the computational complexity of this method for larger systems, compared to the usual methods?
@Ezasur2 жыл бұрын
Loved this one, it seems alot of fun doing the algebra part too!
@aleksaderhansen19456 ай бұрын
Nice video
@molepistol Жыл бұрын
here i am again cute professor........ looks like you gonna be leading me through my second semester exams
@holyshit9222 жыл бұрын
Challenge for you solve it in symmetrical form You solve system in symmetrical form when you solve first order PDE by method of characteristics
@carultch9 ай бұрын
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
@le_nub67112 жыл бұрын
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
@drpeyam2 жыл бұрын
No way, Fourier transform is awful for nonlinear PDE
@joymalya.p2 жыл бұрын
thank you for the nice explanation. Helped me indeed.
@sejr80532 жыл бұрын
Wow, I never knew how useful the laplace function was!!!😆😮
@drpeyam2 жыл бұрын
Right???
@zyctc0002 жыл бұрын
Can we solve this by e to the power of a matrix method too?
@drpeyam2 жыл бұрын
Yes of course, but this is more direct
@geovannycamargo1282 Жыл бұрын
what are the Hilbert transform and the z transform?
@user-yx6is2et5s3 ай бұрын
the only helpful video on youtube but i wonder why this method didn't work for my worksheet
@ojasdeshpande72962 жыл бұрын
I have no Idea what this is XD
@SuperYoonHo2 жыл бұрын
thanks for the awesome video
@drpeyam2 жыл бұрын
You’re welcome 😄
@SuperYoonHo2 жыл бұрын
@@drpeyam 👍
@lego3122 жыл бұрын
Is it possible to differentiate both equations, allowing you to cancel down to second order linear ODE? This seems to give the same answer
@lego3122 жыл бұрын
rather, after differentiation, you can use the derivative to cancel the original eqns.
@xinpingdonohoe39782 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.
@tanjimibrahim664210 ай бұрын
From where we got (s-1)(s-2)
@jacksonmadison99945 ай бұрын
Isn’t Eigenvalue method a lot easier?
@drpeyam5 ай бұрын
This is more direct
@tretyakov31122 жыл бұрын
3:48 how did we get this?
@drpeyam2 жыл бұрын
It’s called Cramer’s rule and there’s a video on that :)
@albiaclifordjanernib.50725 ай бұрын
What is your book reference
@drpeyam5 ай бұрын
I came up with it on my own but maybe consult Boyce diprima?