Question

(ODE) Okay, dealing with finding an interval of solution, something like that for an IVP, full prompt posted below in a moment.

Answer 1

http://i.imgur.com/uwOmtBs.png

Answer 2

I think this is the relevant theorem: http://i.imgur.com/sLoesmC.png

Answer 3

Alright, I'm not really sure what to do with this one, at least on the information given in the book. Reading some more.

Answer 4

So, x obviously can't equal 2 if we keep the coefficients as they are, so the interval of solution can't have 2.

Answer 5

Otherwise, all of the coefficients are continuous on (-infty, infty), so that's satisfied, so I guess the interval is something like,

Answer 6

\[I=(-\infty,2)\cup(2,\infty)\]

Answer 7

But that doesn't sound right, that's too simple, lol, I remember there being two parts to this, the interval of continuity and an interval of validity, and they are not necessarily the same.

Answer 8

The interval I contains the argument of the IVP conditions and is still continous/nonzero coefficients with it, e.g.

Answer 9

\[y(0)=0; \ \ \ [(0)-2]y''+3y(0)=0\], coefficient of lead term is nonzero still

Answer 10

Not sure about this, that union/interval statement.

Answer 11

i like the way you answer your questions on your own ;)

Answer 12

(Lol, I do this for a reason, man. I don't know how to do half of this stuff with confidence, and for the last six hours I've been bumbling through problems with everybody's help, now reviewing everthing I know as quickly as possible and hoping to work through it without mistakes; I work through it here as a contingency in case I need help and need it quickly.)

Answer 13

The thing I lack is confidence or certainty yet.

Answer 14

But I'm not sure if that's right! lol. Checking answers

Answer 15

\[y'' + f(x)y' + g(x)y = h(x)\]
is that the general form of DE in IVP that the existence uniqueness thm referring to ?

Answer 16

I think it's an nth order IVP, that the uniqeness theorem refers to, and this is interesting, the answer doesn't have my top bound, it's not a union

Answer 17

It's just the bottom half of the union, (-infry,2).

Answer 18

i think so, because you want to center the interval about x=0

what exactly does centering the interval mean ?

Answer 19

I'm guessing it's because the bigger interval doesn't contain the arguments of the IVP conditions?

Answer 20

Oh, uh, I dunno, hahahah

Answer 21

le google

Answer 22

Wow, so, uh, that's awful, I did a text search through the book and they literally don't even mention the word in the chapter, let alone section, that "centered" phrase only comes up when talking about power series solutions for obvious reasons and another place very early on where it might be relevant. One sec.

Answer 23

Exact same problem:

Here's something relevant, but the responder starts going off about power series, haven't read it all yet, but it's literally the exact same problem. http://mathhelpboards.com/differential-equations-17/finding-interval-ivp-7326.html

Answer 24

its okay lets move on for now may be... it might become clear after working few problems

Answer 25

Yeah, that barely shows up, I think it's kind of a garbage problem or the textbook went about it poorly, yeah, I agree, posting a new problem shortly.