Using Laplace Transforms to solve Differential Equations ***full example***
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- Опубликовано: 20 мар 2020
- 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?
Done well, really helped me put everything together during these covid self-teaching times.
Thank you! Your videos are so helpful while I'm taking DE online!
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!
Fantastic explanation!
This was magnitudes easier to understand then the way my professor showed it. Thank you
my exam is in a few hours and you are a life saver!!!! thank you!!
I am so grateful I found your channel tata 😭 God bless you!
Thank You Sir , Very Much Helpful Video.
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
Thank you so much sir!
You are the best math teacher ever💥!!!.
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!
this is awesome sir, thank you
Really saving my engineering ass before my midterm thank you :)
Thank you sir 🔥
Thank you this video really helped me !
ilysm
That clarifies a lot! I might not fail now!
Yeah, I hope I will not fail tomorrow
dude be saving math students azzes
Love you sirrrr
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
tnx alot sir.
Where did you get the initial t-shirt with the first and second derivatives on it?
sir post some limit sequense . converge or not. example videos
super new video wow!
God bless your soul.
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?
once you do the inverse laplace, dont you require a Heavside function?
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...
Nice
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
Here i have confusion, how it is 2 b, as we see put -1 as s so it will b -3b
Liked 🙂
How to convert integral to differential by Laplace
Thanks
Thanks so much!!
Woah where can I get that t-shirt!
its in his amazon affiliate shop! a little different but still cool
You could also just plug s =-lnx in
I love you bro
Dr. Bazett, where can i get the shirt? It looks so cool!
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.
👍
where can i get the t shirt your wearing in the start
798//6.10.21
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.
Shirt is kool where can I get it
Where I can get the proof of the Laplace transform of 2nd order derivative ???
Just apply the rule for first derivatives twice in a row
Is that shirt still for sale?
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!
dfkm!
need that tshirt
00:00 nice shirt
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)
i am gay
I would solve that differential equation instead.
laplace makes solving equations with a higher order easier