ODE existence and uniqueness theorem

In this video, I prove the famous Picard-Lindelöf theorem, which states that, if f is Lipschitz, then the ODE y’ = f(y) with a given initial condition always has a unique solution (at least in the local sense). The proof involves some neat analysis; more precisely we use the Banach fixed point theorem, featured in another video of mine. Finally, I’ll give some examples of non uniqueness and nonexistence when the assumptions aren’t satisfied. Enjoy!

[tekaaable]

I take notes, pause, rewind, pause, rewind ... Spending a couple of hours on this vid to get every step of it! Thank you! Im a fresh grad student i math, and I really appriciate this. Best regards from Norway

[JackSPk]

Thank you very much! the context you add while advancing into the proof make it much easier to follow and understand whats happening.

Also your jokes are great! :)

[Rundas]

Integral Equations look very interesting. It would be nice to see something about them from you since I haven't seen anything from the topic of I.E yet and you make everything understandable. So maybe this type of equation would be something for next year :D

[sedeanimugamez]

You are the most epic gamer I love your videos thank you

[danielescotece]

Your videos are helping me a lot with my analysis 2 exam!
Thank you!

[OtherTheDave]

It always makes my chuckle when I hear “x not <whatever>” instead of “x naught <whatever>” .

[MichaelRothwell]

Lovely proof. I haven't looked up the Banach fixed point theorem, but it seems intuitively obvious. Just define xn=T^n(x) and you have a Cauchy sequence (due to the contraction property) and so converges (by completeness) and by a simple limit argument Tx=x (just like for recursively defined sequences of real numbers). From the contraction property we easily show uniqueness. Done!

[StreuB]

I have absolutely no idea what is going on its still badass!!!

[bebarshossny]

i wish you were my professor :, ( i'm so glad we have your videos man

[salvatoregiordano]

Beautiful explanation! You are amazing. Thank you so much!

[ninck]

Great video!
Can you show us why there isn't a similar theorem for PDE?
I've seen an example of an first order linear PDE without any solution and it blew my mind.

[rolfschramek]

Great Video as always. Isn't there a problem defining the metric on X like this, because X is not a vector space, hence not a Banach space. Or maybe I'am misunderstanding something.

[willnewman]

Good video. Does this proof also work for complex differential equations?

[ey]

For x, y belonging to U, mod(x-y) is euclidean distance between them? Since U itself belongs to R^m.Thanks 🙂.

[danielescotece]

Peano and Cauchy, im studying them right now ahahhaha so cool, exactly like a minute ago

[j.v]

Is there a theorem like this for higher order ODE?

[phuocbui]

The final example, i think F need to lipchitz around x0 =1, because F(y(t)) with t around 0 is around y(0)=1, (F(y(0))= F(1)), but you say it lipchitz around 0, can you explain, thank you very much

[shumailaazam]

can we use odes theories in nonlinear pdes? too?

[parthasarker]

Is Lipschitz condition somehow related to the uniform dependence on initial conditions?

[dgrandlapinblanc]

Never in hollidays ? What's a crime ! Thank you very much...