Question

Need some help setting this up, second order non linear DE @ganeshie8

Answer 1

\[m u''+ \gamma u' + k u + \epsilon u^3 =0 \] it's the u^3 term, which I'm not sure about...so I would have thought we needed characteristic equation etc, etc, etc but not sure because of that term.

Answer 2

Maybe \[\mu''+\gamma u' + k u = - \epsilon u^3\] and then it's non - homogenous and solve it from there? Haha, not sure.

Answer 3

homogeneous*

Answer 4

variation of parameters way maybe?

Answer 5

I thought of the method of undetermined coefficients, if that's what you mean?

Answer 6

\[ay''+by+cy=g(x)\] but that u^3 :S

Answer 7

well, that only works for certain g(x) var. of parameters works for any rhs

Answer 8

but hmm... let me think for a few. If someone else comes along please intervene

Answer 9

Hmm I guess we haven't learnt that method, but I'd still be interested in it

Answer 10

http://www.sosmath.com/diffeq/second/variation/variation.html

Answer 11

this may help, I'm sorry but the u^3 is throwing me for a loop in the char eq. http://www.math.umn.edu/~olver/ln_/odq.pdf

Answer 12

I know what you mean, I'll check the links out

Answer 13

I assume, m psy kappa and epsilon are constants?

Answer 14

yup

Answer 15

typical trick is to let \(v=\frac{du}{dt}\)

Answer 16

but not sure on that u^3 hmm

Answer 17

Wait, it's asking to show that the displacement u(t) of the mass from its equilibrium position satisfies the differential equation

Answer 18

wait, it's a plug and chug?

Answer 19

I mean would I still have to find a solution as you would an ODE or just use some physics haha, F=mu''

Answer 20

One sec, let me see if I can get a pic of the quesiton

Answer 21

This is the problem related I found online, way better worded than mine; mine seemed as the initial conditions weren't even part of this part, weird.

Answer 22

maybe it's a plug and chug: \[mu"+\gamma=0\]

Answer 23

ugh, I know I did this before, so frustrating that I forget

Answer 24

Yeah, not sure, I'm reading my book seeing if I can figure it out haha

Answer 25

hey look what I found! http://kaharris.org/teaching/216/Lectures/lec31/lec31.pdf

Answer 26

Ah, I figured that we would have to linearize it, but I'm not sure if I should be using that method haha. Jacobians bleh, maybe Euler's equation or something, reduction of order? Heh

Answer 27

well I mean numerical method isn't out of the question if not a math class, but I'd do it that way on the slides since it is your exact problem

Answer 28

http://www.dcc.ufrj.br/~vitormaia/down/Boyce,DiPrima.ElementaryDifferentialEquations.pdf page 186 in book/ pdf 206 it talks about how to approach it but still not sure how to do it for this question, if you scroll down a bit more the problem will be there to

Answer 29

are u looking for numerical methods?

Answer 30

not for this part

Answer 31

ohh lemme thing hmm jacobians eh

Answer 32

hmm why jacobian, that stuff is used with u have like a certain area, you want integrate over, this would be indefinite integrals

Answer 33

Nooo jacobians

Answer 34

how about power series solutions

Answer 35

Nooo, but I'm willing to look at it xD

Answer 36

oh i just saw ur equation

Answer 37

this is laplace question u dummy

Answer 38

u have u(0) and u'(0)

Answer 39

u shudda said u had those intial values!

Answer 40

<3

Answer 41

omg...I am such na idiot

Answer 42

Lolol, the one I have suggested as it wasn't part of it but in this book it is, and I like this book, so yes xD use the initial conditions!