ODE existence and uniqueness theorem

[@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! :)

[@Rundas69420]

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

[@sedeanimugamez5418]

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

[@OtherTheDave]

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

[@willnewman9783]

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

[@MichaelRothwell1]

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!

[@MichaelRothwell1]

P. S. I came here after watching your great recent video illustrating an ODE with a non-unique solution "because it's not Lipschitz", taken straight from here. BTW, the Lipschitz condition would be equivalent to a bounded derivative for a differentiable function, so it's clear that √x is not Lipschitz for x>0 and so also not Lipschitz for x≥0.

[@ninck8992]

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.

[@StreuB1]

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

[@howmathematicianscreatemat9226]

thank you for applying this term to our science! It felt like coolness wouldn't want to go that far! It made our subject seem boring and dead.

[@rolfschramek7060]

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.

[@danielescotece7144]

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

[@bebarshossny5148]

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

[@SamSarwat90]

How can somebody dislike this....

[@salvatoregiordano9050]

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

[@parthasarker6121]

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

[@drpeyam]

I don’t think so because the Lip condition depends on f, whereas the other one depends more on y0

[@parthasarker6121]

@@drpeyam hey can you give me some resource to study about Lip condition?

[@j.v6334]

Is there a theorem like this for higher order ODE?

[@shumailaazam6]

can we use odes theories in nonlinear pdes? too?

[@ey3796]

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

[@SylComplexDimensional]

CoolFIRE ODE Analysis!!

[@Anders3000]

Great proof, thank you!!

[@phuocbui3347]

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

[@fonzi102]

very well explained, gj

[@gregorymacchio4077]

Thank you!!!

[@shiina_mahiru_9067]

You said we turn it into an integral equation because limit is easier to pass in . . . but . . . did we use any limit in the proof? 😂

[@drpeyam]

Yeah, because the Banach Fixed Point Theorem involves limits 🙂

[@danielescotece7144]

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

[@Jaylooker]

Nice

[@AndDiracisHisProphet]

make it so!

[@dgrandlapinblanc]

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

[@Debg91]

Aahh that's hot!

[@AxiomaticUncertainty]

First

[@deeptochatterjee532]

Good for you buddy

[@floribus5522]

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.