Math 391 Lecture 20 - Series solutions near regular singular points; Intro to the Laplace Transforms

Pay attention to the annotations (added before annotations were discontinued by YouTube). We use a trick to find the indicial equation for second order ODEs at regular singular points, but it only works in one situation--the coefficient of y'' must be x^2. In general, another rule must be followed that is mentioned in the annotations (which you may not be able to see if watching YouTube on your mobile device).

In this lecture, we run through finding series solutions near regular singular points, i.e., finding the series solution to an ODE where the power series are centered at a regular singular point. This semester, this topic is not really in our syllabus, but this is an important idea anyway.

With the remaining time we have left, we introduce the notion of the Laplace Transform and provide some intuition as to where its formula comes from. In the next class, we shall compute the Laplace transforms of some well known functions, and use Laplace transforms to solve linear ODEs.

[swanhtet]

There are many professors who are smart. However, not all of them analyze what sort of methods will make students grasp the material in the most efficient way, like you do. That is why you are my favorite "teacher". #muchrespect

[bselvi]

Literally there's no video on youtube about this topic other than yours. You saved me! After that, I watched 3 more of your videos. Thank you so much! :)

[yehon]

Thank you for the lecture, as mentioned before, you are the only one who seems to cover this topic on YouTube.
You cleared out a few things for me, thanks a lot!

[AkhiDeia]

Have been doing laplace transforms for the last two years and didn't know the actual concept or derivation behind it. Stayed for the last few minutes of the video out of curiosity and everything is crystal clear now!! Why can't other professors be like you ?

[fewdguy]

You're THE man Jhevon. You're one of the handful of people who make YouTube so valuable. Thanks a lot !!

[NickOrloffski]

Greetings from Chico, CA. Your lectures are great and are pretty much the reason I'm passing diffs right now. Thanks for putting them up.

[whoisdanielsong]

Your explanations are great and I will treasure my time watching your lecture! I wish you the best for your teaching career! <3

[Jaikful]

your lectures are really helpful man, thanks for posting them on youtube. Im from Canada and I am using your lectures to study for my diff equations exam

[ownershen]

Wow well explained sir! I had few question in my mind which you straight up answered=) Thank you!

[sharkdavid]

Thanks a lot! Though I wish you covered the case when the indicial roots are repeated or differ by an integer.

[AriHammerfest]

Your lectures are really helpful. Thanks for them)

[MrAHchannel]

why we didn't substitute r=1 in the final answer at 29:54 ?, since we were asked to find the solution for the larger root. x^r would be x

[joseduran]

Are there any videos on the eigenvalues and eigenvectors

[randomhandle]

I'm struggling to understand why your method for finding the associated Euler equation works though. It seems to me it doesn't work for this example:

(1 − x^2)y'' − 2xy' + k(k+1)y = 0 (k is a constant) (This is Legendre's equation)

My book describes the associated Euler equation as

r(r-1) + p0 * r + q0 = 0 where p0 and q0 are the limits you get when finding if a point is a regular singular point.

Anyway, your lectures are great. I wish you the best!

[Thegtrs]

11:28 why multiply by x? Do you always prioritize the first term? If you left it as is you would get a value of -r+3=0.

[oskarvedin]

46:30 Isn't the recurrence relation only valid for n≥2? When you solve for a_(1) you get an a_(-1) term in the numerator?

[TheBrucekee]

I love your videos! A huge help through the little uncertain steps. I have a specific question about change of indices. Is it possible to have n = -1 for the summation? would we just set the index equal to zero (0)?
Thanks again from UCLA =]

[iftikharakhan]

Great Lecture as always. Thanks! ( Waiting for pizza )