Question

ODE question again. So I need to determine for what r values the ode has a solution of the form \(y=t^r\) for t>0. What method yields only \(t^r\)? I know ce^rt, but not one for t and cannot locate it in the book.

Answer 1

@ganeshie8 @Marki

Answer 2

the actual problem, I can do, but I would use an integrating factor for this problem not whatever method this is

Answer 3

so for matrices or equations ?

Answer 4

it's an equation, specifically:\[y''+4t^{-1}y'+2t^{-2}y=0\]

Answer 5

I am assuming there is some method similar to the characteristic eq. then solve for r and have a generic answer

Answer 6

this is special equation is it of Legendre or Bessel i dont memorize these stuff ;-;

Answer 7

no clue, We haven't learned it. This is something for the first week's homework...

Answer 8

ok i wanna refresh m mind what topics u have for the HW ?

Answer 9

uhm, well a lot there is like 50 problems, so first was, convert to canonical form, second verify something is the solution of the de, third determine r values which give an answer of ce^rt, fourth, this stuff with the t^r, fifth all of the numerical methods, sixth translate the dif eq tot he origin, seventh method of successive approximations, eighth sequence convergence(idk i haven't tried those yet)

Answer 10

ok i'll try something with Bessel

Answer 11

ok, and can you link to something, I've never heard of him

Answer 12

check equation 2
http://mathworld.wolfram.com/BesselDifferentialEquation.html

Answer 13

interesting @wio

Answer 14

sowe should convert ur equation to this form
\(y''+t^{-1}y'+(1-n^2y^{-2})y=0\)

Answer 15

I like to guess, because I've forgotten a lot\[
y=t^{r}\\ y' = rt^{r-1}\\y'' = r(r-1)t^{r-2}
\]And then: \[
r(r-1)t^{r-2} +(4t^{-1})rt^{r-1} + (2t^{-2})t^r=0
\]

Answer 16

@Marki that seems a little complicated, this is the first week... I would need some explanations in english. I'm a little confused by the terminology.

@wio cool it looks like a derivation

Answer 17

ok then forget it :P
this is the only way i remember tbh for this equation
idk maybe @ganeshie8 could also know

Answer 18

Hmm....\[
\bigg(r(r-1)+4r+2\bigg)t^{r-2} = 0t^{r-2}\implies r^2+3r+2=0
\]

Answer 19

lol it's cool,

Answer 20

so I guess to use that we would need an n, and to +1 and -1?

Answer 21

Now we can verify whatever solution we get.

Answer 22

@wio, I like your work, but I am confused a little let me reread it

Answer 23

ok, so that is awesome wio! I get it! and I think it will work perfectly!

Answer 24

What is the method you are supposed to be using?

Answer 25

no clue

Answer 26

lol that's the issue

Answer 27

the only thing similar is the method where you solve the char eq then have a solution of the form ce^rt, but the book doesn't really cover that either

Answer 28

this teacher is sort of a discombobulated, barely speaks english mess

Answer 29

Are you supposed to put it in canonical form?

Answer 30

lol I wish, but no not on this one. That would have been much simpler

Answer 31

Okay, I think that doing what I did is the answer then?

Answer 32

if it helps the original eq is this, \[t^2y''+4ty'+2y=0\]

Answer 33

I think it's possible

Answer 34

I have convinced myself that you are 100% correct wio, thanks, really really great reasoning skills!!!

Answer 35

ok here is wolfram solution

Answer 36

but term of x instead of t xD
save it u might use it some where xd

Answer 37

that's awesome, it is wio's method!