Using Laplace Transforms to solve Differential Equations ***full example***
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- Опубліковано 29 сер 2024
- How can we use the Laplace Transform to solve an Initial Value Problem (IVP) consisting of an ODE together with initial conditions? in this video we do a full walkthrough beginning with the differential equation, converting it to an algebraic equation via the Laplace Transform, solving that algebraic equation, and finally converting back to a solution to the IVP through the Inverse Laplace Transform.
This is part of my series on the Laplace Transforms in my Differential Equations Playlist: • Laplace Transforms and...
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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
both are simple ways ... it depends upon us to choose which way to use
@@10harinims61 I guess it depends on what you are used to yeah
@@10harinims61 but one is simpler, you can choose the harder way if you want lol
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
Thank you so much I got so stuck because I didn't understand his method at all
You just explained how to do this ten times better than my college professor did earlier today. Thank you for the help!
bold of you to get clarification early instead of cramming before a test
@@balls4924 sup brah, my final is tomorrow 💀
ight i think im getting a 75-80. was way easier than previous years but still fucked up some questions
@@nerd2544 Any update?
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.
If you found your answer them pls refer me source too! I seriously want to know
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.
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.
This is something I’m curious about just learning about them this week and am curious what the intuition is behind them
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!
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!)
This is one of the best maths videos ever watches, many thanks.
Glad it was helpful!
This was magnitudes easier to understand then the way my professor showed it. Thank you
Very well made and clear. Thank you.
my exam is in a few hours and you are a life saver!!!! thank you!!
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
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!
I am so grateful I found your channel tata 😭 God bless you!
Done well, really helped me put everything together during these covid self-teaching times.
Really saving my engineering ass before my midterm thank you :)
dude be saving math students azzes
Thank you! Your videos are so helpful while I'm taking DE online!
That clarifies a lot! I might not fail now!
Yeah, I hope I will not fail tomorrow
You are so polite i wish having happienes in your live
this is awesome sir, thank you
You are the best math teacher ever💥!!!.
Fantastic explanation!
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...
Thank you sir 🔥
Thank you so much sir!
Great video! How to du know that L{y"}= s^2y(s)-sy(0)-y´(0)? Is there any intuitiv way to see this?
I walk through this in an earlier video in the Laplace playlist:D
Thanks
sir post some limit sequense . converge or not. example videos
Woah where can I get that t-shirt!
its in his amazon affiliate shop! a little different but still cool
Thank You Sir , Very Much Helpful Video.
Love you sirrrr
God bless your soul.
once you do the inverse laplace, dont you require a Heavside function?
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?
super new video wow!
Here i have confusion, how it is 2 b, as we see put -1 as s so it will b -3b
Nice
Thank you this video really helped me !
How to convert integral to differential by Laplace
ilysm
Liked 🙂
I would solve that differential equation instead.
laplace makes solving equations with a higher order easier
Where did you get the initial t-shirt with the first and second derivatives on it?
tnx alot sir.
In which case of differential equation I can't apply the Laplace transform? Or can I apply Laplace everytime I want?
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.
You could also just plug s =-lnx in
Where I can get the proof of the Laplace transform of 2nd order derivative ???
I love you bro
Shirt is kool where can I get it
798//6.10.21
Dr. Bazett, where can i get the shirt? It looks so cool!
where can i get the t shirt your wearing in the start
Can a Laplace Transform be used in a boundary value problem?
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)
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.
👍
Damn, this's hard. What level of math is this recommended for?
Differential equations
it isnt hard ... dont give up ... keep trying... try to get the basic concepts ... u will definitely find maths easy
@@mathadventuress Yep! Diff-EQ is diff-e-cult!
00:00 nice shirt
Is that shirt still for sale?
need that tshirt
dfkm!
i am gay
why are u happy ... im mad bcoz of that im offended
8:08 that equivalency statement doesn't provide any insight
inverse laplace of transform of F(s) is f(t) right
the same way laplace inverse of Y(s) is y(t)
Thanks
Thanks so much!!