The Big Theorem of Differential Equations: Existence & Uniqueness

The theory of differential equations works because of a class of theorems called existence and uniqueness theorems. They tell us, given an initial value problem, when that IVP has a solution, and if a solution exists, then it also tells us it is unique. We will focus on the Picard or Picard-Lindelof existence and uniqueness theorem which applies to first order differential equation. A key focus will be on the domain on which a unique solution is guaranteed to exist.

0:00 Intro
1:37 Ex: Existence Failing
4:10 Ex: Uniqueness Failing
9:15 Existence & Uniqueness Theorem

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[ThabetMarwa]

I knew this theorem but always thought of it in the cases where both the function and its derivative were continuous. It’s beautiful to see the exceptions explored

[benjamindilorenzo]

This is so good: its just that every video i saw, i understood almost everything at the first time watching. And thats a lot, given that you actually condense a lot on information into small windows of time.

Thanks for giving that free lectures. Your sparkling excitement for mathematical patterns and thinking is really refreshing and transfers allready important factors to good learning.

[arsenalaman]

After a long time I hear you sir actually I love your concept soo I always see your videos as I'm from India

[georgesadler]

Professor Bazett, this is an outstanding video/lecture on The Big Theorem of Existence and Uniqueness in Differential Equations. These two theorems build a solid foundation for understanding Differential/Partial Differential Equations.

[kevincarvalhodejesus]

Dr.Trefor, thank you so much! I like your explanations along with the graphics. that makes it all clearer

[pipertripp]

I'm glad I'm seeing these as you put them out rather than having to binge watch them at some time in the future. I'm working on numerical methods right now, but ODEs are really neat and it's nice to have these videos to keep them close to the surface, so to speak.

[amichair]

Trefor, you are a god of learning on youtube. Kudos!

[yugalteotia]

I am following your discrete maths playlist for 3-4 days, those explanations are awesome. But I am worried why your content isn't much popular even though the quality is too good.🔥🔥
Hope I score well in exams🤞🏻.

[parmodkumarph.d.]

Thank you Dr Trefor Bazett for these videos! Please also discuss some real life applications of these functions and how we can form a research problem further on these!

[jakemol]

you are saving an engineer in training. This is the most intuitive explanation ever. Thankyou.

[HosRo]

You are such a good teacher! I love the emphasis and the animated style of teaching! Very conducive to learning. Thank you!! :)

[maestro_]

Thanks Alot Sir For The Link To The Geogebra Slope Field Grapher!, Very Nice and Useful

[gymnasiematematikmedigor]

One thing is unclear to me.
In the example of y’=1/x it is easy to see that the solution does not exist since 1/x is undefined at 0.
But can we have a differential equation y’=f(x, y), where f is perfectly defined at a point, BUT the solution through that point does not exist? I hope you could give an example.
Thanks a lot for a great video!

[chg]

What a fantastic interpretation!!! Thank

[the_eternal_student]

That part about partial derivatives proving uniqueness was interesting.

[sandeepchauhan]

You are doing a great work a great help sharing knowledge is the best charity

[annali]

this is just so clear and I like the way you talk, appreciate

[tolosanigussie]

Thank you Doctor, your explanation is clear and smart. But I didn't understood that how y=±(2/3 x)^(3/2)

[debiprasad]

You are absolutely an amazing teacher
But sometimes you are just so fast to be understandable ;)

[MinhLe-kccw]

Thank you for your video! That's really informative and logical