the outstanding Laplace method for solving systems of ode

[@knowitall6677]

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?

[@factsheet4930]

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]

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

[@mujimuji6002]

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!

[@pheldge8806]

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 :)

[@arturb2996]

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.

[@MrFlaviojosefus]

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

[@marccox8977]

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-dl8nk5bf8v]

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]

Right, I mention eigenvalues at the end

[@DarkMonolth]

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]

Yes exactly!

[@DarkMonolth]

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]

Of course!!

[@NittyLittyNiturzion]

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

[@brendanlawlor2214]

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

[@BigAsciiHappyStar]

Beautiful video!!! 😄😄😄

[@vaughanwilliamson173]

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]

My lecture notes :)

[@drpeyam]

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

[@vaughanwilliamson173]

​@@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?

[@CornishMiner]

So cool. 😎

[@MushookieMan]

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

[@Ezasur]

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

[@aleksaderhansen1945]

Nice video

[@molepistol]

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

[@holyshit922]

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

[@carultch]

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_nub6711]

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]

No way, Fourier transform is awful for nonlinear PDE

[@joymalya.p]

thank you for the nice explanation. Helped me indeed.

[@sejr8053]

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

[@drpeyam]

Right???

[@zyctc000]

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

[@drpeyam]

Yes of course, but this is more direct

[@geovannycamargo1282]

what are the Hilbert transform and the z transform?

[@user-yx6is2et5s]

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

[@ojasdeshpande7296]

I have no Idea what this is XD

[@SuperYoonHo]

thanks for the awesome video

[@drpeyam]

You’re welcome 😄

[@SuperYoonHo]

@@drpeyam 👍

[@lego312]

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]

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

[@xinpingdonohoe3978]

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.

[@tanjimibrahim6642]

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

[@jacksonmadison9994]

Isn’t Eigenvalue method a lot easier?

[@drpeyam]

This is more direct

[@tretyakov3112]

3:48 how did we get this?

[@drpeyam]

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

[@albiaclifordjanernib.5072]

What is your book reference

[@drpeyam]

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

[@Enigma758]

"Beautiful L" :)

[@aliexpress.official]

That is waaaay too much work for a linear system

[@drpeyam]

Not really?

[@Re-lx1md]

"Beautiful L" 😂

[@Hectorreyes-]

i cannot take u seriously my frend

[@drpeyam]

Ok

[@LuisBorja1981]

Your beautiful L looks a lot like a fancy Z