Existence and Uniqueness of Solutions (Differential Equations 11)

THIS VIDEO CAN SEEM VERY DECEIVING REGARDING CONTINUITY. As I watched this back, after I edited it of course, I noticed that I mentioned continuity is not possible at Endpoints. This is NOT true, as we can consider one-sided limits. What I MEANT was that since we must consider an OPEN INTERVAL of continuity in the NEIGHBORHOOD of our point, the points we are checking must NOT be at endpoints of continuity. If they are, a neighborhood around those points is not possible. I failed to make a distinction between continuity and continuity in the NEIGHBORHOOD around a point. I am TERRIBLY sorry about the deception. So, one more time; continuity IS possible at an endpoint, but the points we are checking must show continuity on an OPEN interval of continuity to satisfy the NEIGHBORHOOD around the point. Therefore, if a point we are checking falls on the endpoint of continuity, this fails our theorem. When I refer to Continuity around the point, I mean exactly that... the neighborhood around that point, not just continuity of the function, but continuity on a open interval around the initial value. I know that I make reference to that EXACT statement several times in the video, but it was not clear enough in the beginning of the video. I assumed that the distinction would be understood and that my usage of Continuity would be limited to the Continuity of the function in the neighborhood of the point we are CHECKING, and that was my mistake. I tried to make it understandable and made a fundamental error. When I refer to "the function is not continuous at an endpoint", my meaning is that if the INITIAL VALUE is at the endpoint of an interval of continuity, we fail to have continuity on an OPEN INTERVAL.... a NEIGHBORHOOD around that point. Continuity HERE is an "open" idea because of the necessity of continuity in the NEIGHBORHOOD around our point.. My apologies for any misunderstandings this may have caused. In my mind, as I was filming this, it was clear but just keep in mind that when we refer to continuity here, it's continuity on an OPEN interval around the POINT WE ARE CHECKING. Hope this helps.

How to determine existence of solutions to Differential Equations and when those solutions will be Unique.

[ProfessorLeonard]

THIS VIDEO CAN SEEM VERY DECEIVING REGARDING CONTINUITY. As I watched this back, after I edited it of course, I noticed that I mentioned continuity is not possible at Endpoints. This is NOT true, as we can consider one-sided limits. What I MEANT was that since we must consider an OPEN INTERVAL of continuity in the NEIGHBORHOOD of our point, the points we are checking must NOT be at endpoints of continuity. If they are, a neighborhood around those points is not possible. I failed to make a distinction between continuity and continuity in the NEIGHBORHOOD around a point. I am TERRIBLY sorry about the deception. So, one more time; continuity IS possible at an endpoint, but the points we are checking must show continuity on an OPEN interval of continuity to satisfy the NEIGHBORHOOD around the point. Therefore, if a point we are checking falls on the endpoint of continuity, this fails our theorem. When I refer to Continuity around the point, I mean exactly that... the neighborhood around that point, not just continuity of the function, but continuity on a open interval around the initial value. I know that I make reference to that EXACT statement several times in the video, but it was not clear enough in the beginning of the video. I assumed that the distinction would be understood and that my usage of Continuity would be limited to the Continuity of the function in the neighborhood of the point we are CHECKING, and that was my mistake. I tried to make it understandable and made a fundamental error. When I refer to "the function is not continuous at an endpoint", my meaning is that if the INITIAL VALUE is at the endpoint of an interval of continuity, we fail to have continuity on an OPEN INTERVAL.... a NEIGHBORHOOD around that point. Continuity HERE is an "open" idea because of the necessity of continuity in the NEIGHBORHOOD around our point.. My apologies for any misunderstandings this may have caused. In my mind, as I was filming this, it was clear but just keep in mind that when we refer to continuity here, it's continuity on an OPEN interval around the POINT WE ARE CHECKING. Hope this helps.

[keenanfitzharris]

Hey Prof Leonard! I'm currently an undergraduate engineering student and your videos have helped me passed most of my math courses. A huge thank you. I now plan to continue on the graduate school to pursue a career as a professor as well. You have inspired me and I am sure many other students that have watched your videos. Your style of teaching is one I plan to adopt as I continue to teach other people. Thank you for all that you do!

[theclimb]

I appreciate that you continuously repeat concepts before moving on. It helps me better grasp the material.

[maxanderson]

It is incredible the difference in teaching can make on your comprehension of material. I'm at a top ranked college worldwide and my professors make me feel stupid. 10 minutes into this video it makes perfect sense.

[Bigpaa]

Finally someone who can explain Existence and Uniqueness thoroughly and completely! It's really not that hard once it clicks.... it's just getting the brain to understand lol

[shensley]

So glad i waited an extra year to take this course. Thank you Prof!! You've made my life sooo much easier.

[ronaljabali]

I don't know how can we pay back your favors, overseas we are watching your videos, learning from you, talking to you if front of screen and mastering our courses. thank you in advance my best teacher ever.

[gallonsofwater]

Holy moly, do you know how many hours I've spent rereading 2 pages of a book? And this video manages to make me understand all of that formal jargon in less than an hour. As always, thanks for your service sir!

[leighkong]

Thank you Professor Leonard! This series helped me understand and organize everything I know and everything I don't know about differential equations.

[Eleni_E]

My DiffEq professor is a lot better than many I've had and she chose a decent enough book, but the fact that her course is at 8am means that I'm not always awake enough for the best notes... so, as always, thank you for making these!!

[shahzadnawazmalik]

You are really determined to teach by heart.:)

[chillproductive]

Thank you for helping students pass their exam! You for shure helped me out a lot!!

[abagel]

OMG!! This saved my grade! Thank you so much!

[sanab]

Prof. Leonard is the best math teacher!

[brethagen]

Trying to condense this down to simple rules. May come back and edit this.

1. Plugging the point(s) into the original equation will tell you whether a solution exist.
2. Taking the partial w/ respect to the dependent variable will tell you whether the solution is unique.
3. If the function fails the uniqueness test, that means there is one more more solutions.

[ericaredman]

Thank you soooo much for posting these videos! My professor has a heavy accent and is really hard to understand. You're my savior!

[josephchen]

God bless your soul. This was so helpful!!!

[iancoleman]

"just be open minded, man" *fart noise* "that was the board"

classic

[busraozkan]

I ALWAYS WATCH THESE WONDERFUL MATHS VIDEO. WHAT A GREAT PROFESSOR . GOD BLESS YOU SIR FOR THE KNOWLEDGE YOU GIVE US. WISHING YOU JOY

[johnhocutt]

Hello, Prof Leonard,

Question: Doesn't the existence of a unique solution imply that there is only one solution? How could there be two (or more) unique solutions at a point? If there's more than one solution through a point, then the solution is not unique, yes?

On a personal note: I'm a professional math and science teacher & tutor. I've followed your work for years. The only other Internets serious math teacher(s) I would compare you to is Gil Strang and Art Mattuck out of MIT. All of you are rock solid in the craft. Even as good as they are at not making large conceptual jumps in solutions/derivations, you're even better at making sure that students don't loose the forest because they get blocked by a tree (i.e., an arithmetic/algebra misunderstanding derails the student's being able to see what is going on.) You are one of the few people who is a master of both the content and the pedagogy which makes you so effective. I like to think we're kindred spirits and I offer you my admiration and thanks for what you've generously shared with math students everywhere. Be well, my friend! -jlh