How to solve ODEs with infinite series | Intro & Easiest Example: y'=y

In this video we see how to find series solutions to solve ordinary differential equations. This is an incredibly powerful tool that really expands the amount of differential equations we can solve. We will first introduce power series and their basic principles, along with the standard series expandions for sin(x), cos(x), e^x and 1/(1-x), the geometric series. We'll then see how to substitute the series solution into the differential equation, manipulate indices, and get a recurrence relation on the coefficients. By studying this recurrence relation we can recover the familiar answer to y'=y that has a solution of e^x.

This video is part of a Differential Equations playlist, so go there to check out the next video on series solutions which dives a bit more into the theory behind this.

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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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[vinithadaniel]

I dont generally like a video... But I cant navigate away from this page without liking it... What an excellent explanation! I have never thought about, why, or why series solution... This video really opened my eyes... Million thanks!!!

[saikatify]

This was really neat and like the opposite of generating functions !
We used generating functions to take a recurrence relation and then attach it to an infinite power series to solve it.
Here we start with an infinite power series and finish with a recurrence relation !

[chawnneal]

Thank you so much, I was so excited for this episode!
I see series pop up so much in higher mathematics that I wish it wasn't just introduced as a popup in Calc 2!
Really seeing the full stack of theory to example makes this topic so much more useable! Thank you!

[jjosesillo]

man how can i thank u... now i dont have money but after my exams im going to goin as a member cause u deserved it !

[hamizoo]

This clarified the one part I don't like and often confuse in differential equations. Thanks a lot sir

[caosspearbr]

Your videos make me want to learn everything!! It's awesome!!!

[georgesadler]

Professor Bazett, thank you for an outstanding video/lecture on How to solve Ordinary Differential Equations with Infinite Series. This topic gets messy/problematic with higher order derivatives. Pattern recognition is also. important when solving problems in Mathematics.

[dariorodriguez]

Prof. Bazett, you are such an amazing instructor, I owe you my A on Math Methods course :)

[pipertripp]

Great stuff. It's crazy how often series crop up in the study of ODEs. Definitely a subject that I need to review in great detail as it's the area of Calc 2 that I have always been quite weak in. I'm not quite sure why that is, but I've always struggled with series. I'll be correcting that in the coming months for sure. Trefor, in your experience as a math prof have you found that your second semester calc students struggle more with series that other subjects in Calc 2? If so, why do you think that is? If not, what seems to be the most challenging subject for your students?

[PhysicsBanglaTutorial]

Thanks a lot dear sir❤. Watching from Bangladesh 🇧🇩

[manrajmann]

Suddenly trimmed bread surprised me as a proceeded from previous video !

[HungDuong-dtlg]

What you showed from using the Ratio Test was that the series converged for all x. However, the Ratio Test did not guarantee that it indeed converged to the function e^x. I think in order to prove the series converges to e^x, you need to show the error E_n(x) = e^x -P_n(x), where P_n(x) is Taylor polynomial of degree n, goes to 0 as n goes to infinity. Is it correct? I might be wrong.

[mayoussaghennam]

Wooow Amazing! Any videos about PDEs sir??

[FayçalDaoudi-nz]

Which book or books are the best for learning to write proofs? (your opinion). As always thanks for everything!

[ronycb]

The Test Fails means it is Inconclusive. I was wondering what it meant for a long time.. Thanks.. ❤

[physicslover]

Sir you covered very well the concepts of trigonometric integration including the powers of sin and cosine will all possible combination but sir please make a video on trigonometric integration of tan and secant . There is one case in this kind of trig integration of tan amd secant which is not discussed in any of the books of calculus I have read so far. This case is when the power of tan is even (not 2) and the power of secant is odd (not 1). I have struggled so hard to solve this problem but I can't manage to do I need your help to integrate
Tan²*²(x)sec³(x)dx
Note here tan is raised to the fourth power.

[gentlemandude]

4:59 - I thought you said something else entirely here.

[mishudhar]

Firist view. Just missed this power series method question in yesterday's exam.

[jamesnguyen]

another good video as always 👍 thank you

[Caleepo]

Hello, prof. Will you be making a series in about DE covering seperable, linear, homogenuous, ... DE’s ?