Question

@ganeshie8

Answer 1

ganesh is busy :)

Answer 2

Ganeshieeee I need help understanding something, I'm doing the method of undetermined coefficients for non - homogenous equations.

So I have \[y''-2y'+y=te^t+4\] as my second order ode

What I don't understand is the \[y_p(t)\] part, let me just show you...haha

Answer 3

Haha, it's cool, you can help me :D

Answer 4

as shultz would say: i know NOTH_THING!!

Answer 5

So I first find the non - homogenous equation, which gives me the fundamental set of \[\huge (e^{R_1t},te^{R_1t})\] as there is one simple solution, so my characteristic equation is \[y(t)=c_1 e^{R_1t}+c_2te^{R_1t}+y_p(t)\] it's actually the +4 part, so would my \[y_p(t) = e^tA_0t+4?\]

Answer 6

y''-2y'+y = t e^t+4

y''-2y'+y = 0

yh = e^(t) (c_1+ c_2 t)

yp = e^(t) (f(t)+ t g(t))

Answer 7

first fine the homogenous solution ... yes

Answer 8

instead of constants, let c_1 and c_2 be some function of t

yp = e^(t) (f(t)+ t g(t))

insert it and solve for the nonhomegenous solution

Answer 9

My notes are pretty confusing for this...hmm ok let me see because it states \[y_p(x) = x^se^{\alpha x} (A_0+A_1x+...+A_nx^n)\] where S and alpha determine the type of solution

Answer 10

yp is not 4

Answer 11

Yeah I meant to write +4 is where it's confusing haha, and sorry about the tiny font in the last part

Answer 12

I think I do understand the process, just \[y_p(x)\] I'm a bit confused

Answer 13

\[y_p = \color{red}{\cancel{e^{t}~(f+ t~g)}}\]
\[-2y'_p = \color{red}{\cancel{-2e^{t}~(f+ t~g)}}\color{green}{\cancel{-2ge^{t}}}-2e^{t}~\underbrace{(f'+tg')}_{=0(why?)}\]
\[y''_p = \color{red}{\cancel{e^{t}~(f+ t~g)}}+\color{green}{\cancel{ge^{t}}}+e^t~(f'+ t~g')]+\color{green}{\cancel{ge^{t}}}+g'e^{t}\]
\[=t~e^t+4\]
-------------
\[-2e^t(f'+tg')=0\\
e^t(f'+tg')+g'e^t=te^t+4\]
-----------------
\[f'+tg'=0\\
f'+g'(t+1)+g'=t+4e^{-t}\]
solve f' and g' to determine f and g

Answer 14

+g' was meant to be deleted, but forgot it in the process of trying to code it nicely:/

Answer 15

f' = -tg'

-tg' +tg' +g' = t+4e^(-t)

g' = t+4e^(-t)
f' = t^2+4te^(-t)

Answer 16

bah, f' = -(t^2 + 4t e^(-t))

Answer 17

Haha, well the way I learnt, I'm going to set it up, I think I do follow yours..so \[y_p(t) = t^1e^{t}(A_0+A_1t)\] and then I solve for \[A_0, A_1t\] by taking the derivative and etc

Answer 18

Which is the same as solving for f and g in yours

Answer 19

correct

Answer 20

Oooook cool, so I noticed I have to pay very close attention to degree of t and alpha as I presented above

Answer 21

Which helps set up yp

Answer 22

The 4 seemed a bit strange, that's what got me

Answer 23

its just part of the diffyq to start with, ive never tried to guess ahead as to what may transpire, at least not with any degree of accuracy :)

Answer 24

Haha, that sounds good, but I think I should know when I have less than an hour for midterms xD

Answer 25

ive been binge watching doctor who, just in case i need to do exactly that :)

Answer 26

you're also using superposition right

Answer 27

Yup

Answer 28

But I think that has to do with the wronskian...

Answer 29

I suck with math theory lol

Answer 30

I think you might be doing this way already, but stating it here anyways as im not so sure :

y''-2y'+y=te^t+ cost + 4

directly using the method of undetermined coefficients is going to be a nightmare for above equation

instead, we can use superposition :
1) split the equation :
y''-2y'+y=te^t
y''-2y'+y=cost
y''-2y'+y= 4

2) find a particular solution for each equation using undetermined coeff thingy

3) add up all the solutions

Answer 31

Ooh, no we never did that

Answer 32

then you will feel amazed using it for the first time
it saves lot of donkey work when the function on right hand side is long

Answer 33

So basically you're solving 3 non homogenous equations, I could see how that's probably a bit faster

Answer 34

solving 3 independent equations is always faster than solving a system of 3 dependent equations

Answer 35

Ok nice, because not all of them require undetermined coeff right?

Answer 36

oh nvm

Answer 37

you wont know the power of superposition unless you use it few times

Answer 38

its not a good idea to learn new stuff 1 hour before the midterm, so maybe save that for later :)

if you ever take linear circuits, you will get a chance to see the beauty of superposition in fullest !

Answer 39

Haha, well I was just stating because we have midterms that are 50 minutes long xD, it's not for a while

Answer 40

I'm just working on an assignment, as all my questions are based on them

Answer 41

Ahh then you should definitely learn how to use superposition with undetermined coefficients

Answer 42

Alright cool, I'll definitely look into it, I also have a question that requires variation of parameters where you have to use reduction of order, which is going to suck

Answer 43

Oh and a linear ode with y^(6) omg xD

Answer 44

Ok thanks ganeshie and amistre

Answer 45

looks you're having lots of fun ;) good luck !

Answer 46

Yeah..great way to spend the weekend that's for sure >_>

Answer 47

@amistre64 are you watching season9 it on tv ? we don't get BBC in our present dish plan
i started watching doctor who couple of weeks back
completed first 3 seasons, about to start 4th season...

Answer 48

Haha, nice, I've only seen some of season 1, and now it's already on season 9...I'll have to catch up in the summer

Answer 49

first 3 seasons are not so great, none of the characters are likeable so far... at least to me..
i guess im watching it because some episodes are good and it is a nice science fiction

Answer 50

If you want to watch some awesome sciencefiction and just cool fiction in general, check out the twilight zone, you will definitely not regret it, a lot of shows including Dr. Who have copied it over time, the original (black and white) is amazing.

Answer 51

do you know any nice site to watch these for free ?
i have been watching here
http://www.couchtuner.la/watch-doctor-who-online-11/

but this site doesn't seem to have twilight zone

Answer 52

Try http://thewatchseries.to/serie/the_twilight_zone

Answer 53

Nope. most of the links to videos are broken in that site

Answer 54

Ah, well first episode I tried works, you can probably just google "watch .... free online" and a lot of links will come

Answer 55

Of course this is all legal *cough*

Answer 56

@ganeshie8 i started watching it back when i was like 10 years old ... so ive been catching up on the classic series :)

Answer 57

first and second doctors have alot of missing episodes that are lost, been using .... dailymotion for the most part, and some hulu from my sisters account

Answer 58

Doctor who is on netflix, not sure if it has all seasons though

Answer 59

Ah yes, all seasons are on it! Nice!

Answer 60

wasnt sure how id like the new doctors, 10 and 11 ... but after watching them they turned out pretty awesome. still like the classics tho

Answer 61

oh so the doctor changes in each season is it ?
over the first four seasons, doctor has changed only one time... im not liking this doctor, so can't wat to finish this season to see a fresh face lol

http://www.couchtuner.la/watch-doctor-who-online-11/
above link is very good, i watched every episode in frist 3 seasons from that site w/o any problems

Answer 62

Have you seen the twilight zone ever amistre?

Answer 63

the doctor 'regenerates' when his lifespan/contract requires it :)

Answer 64

used to watch the old black and white TZs

Answer 65

Hehe yes, it's amazing

Answer 66

the first doctor back in teh 60s was old and ill/dyeing, so they decided to go for a 'regeneration' ability .... allows new actors and adventures to expand the series with

Answer 67

Did not know that, you sir are a true whovian

Answer 68

lol, got noting better to do to pass the time :)

Answer 69

i think i recall a twilight zone, a bogey man under this kids bed was stealing/killing kids at night. but his rule was that he could not harm the kid whose bed he slept under.

one night while at the park, the kid is talking to a bogeyman, and at the end you find out that it is one from a different kids bed ....

Answer 70

Ohhh yeah, I remember that one, that was in later years I think from the 80s, the stories they would come up with...

Answer 71

I think you can find some of these episodes on youtube

Answer 72

you can, and i believe alfred hitchcoc.k was in the same genre

Answer 73

For sure, also a lot of actors were involved in TTZ

Answer 74

1212. im headed out ... have fun :)

Answer 75

Later, thanks for the help