Perturbation method for Duffing-type equation with cubic nonlinearity

[Johnmm]

Hi everyone,

I need some help to solve this differential equation.

The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system:

diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 "

Currently, I am still reading and studying on this perturbation method.

Any advise or directions of what I need to do is much appreciated.
Thank you for your help in advance.

John

 

[JJacquelin]

Hi !

First, solve the non-perturbated ODE.
Then, add a term with coefficient epsilon (see attachment) :

 

[Johnmm]

Hi !

First, solve the non-perturbated ODE.
Then, add a term with coefficient epsilon (see attachment) :

Hi JJacquelin

Thank you very much for your help,
I will try it and see if I could reach the result in that picture.
Thank you again for helping me out.

 

[Johnmm]

Hi !

First, solve the non-perturbated ODE.
Then, add a term with coefficient epsilon (see attachment) :

Hi JJacquelin,

Line 7 on your attached picture, I had a little different than yours. I had X^3 instead of X^2.
I am still working on it. Please let me know.
Thank you,

 

[JJacquelin]

Hi Jhonmm !

You are right it isn't X^2 but X^3. This changes some terms of my comments, but this doesn't change the method to solve the problem. I suppose that you can continue by yourself.

 

[Johnmm]

Hi Jhonmm !

You are right it isn't X^2 but X^3. This changes some terms of my comments, but this doesn't change the method to solve the problem. I suppose that you can continue by yourself.

Hi JJacquelin,
Thank you for the help.
I will follow this method to the rest.

 

[HallsofIvy]

"Perturbated"? Isn't the standard "perturbed" simpler and better?

 

[Johnmm]

"Perturbated"? Isn't the standard "perturbed" simpler and better?

I think it is perturbed.

 

[Strum]

You should probably give us some boundary conditions since otherwise you might generate solutions growing without bound.

 

[Johnmm]

You should probably give us some boundary conditions since otherwise you might generate solutions growing without bound.

The question didn't give boundary conditions. But I think it should be x(0)=x₀ and $\dot{x(0)}$=$\dot{x}$₀.

 

[Strum]

Maybe take a look at http://en.wikipedia.org/wiki/Poincaré–Lindstedt_method. Or you could expand using 2 time scales instead using for example
$$
\frac{d^{2}x}{dt^{2}} = \partial_{\tau\tau}x_{0} + \epsilon\left( \partial_{\tau\tau}x_{1} + 2\partial_{\tau T}x_{0} \right)
$$
Differential equations with oscillating solutions doesn't always behave nice when using the usual perturbative approach.

Edit: The wiki link actually treats the same equation as you are looking at! Sometimes you're lucky I guess.

 

[Johnmm]

Maybe take a look at http://en.wikipedia.org/wiki/Poincaré–Lindstedt_method. Or you could expand using 2 time scales instead using for example
$$
\frac{d^{2}x}{dt^{2}} = \partial_{\tau\tau}x_{0} + \epsilon\left( \partial_{\tau\tau}x_{1} + 2\partial_{\tau T}x_{0} \right)
$$
Differential equations with oscillating solutions doesn't always behave nice when using the usual perturbative approach.

Edit: The wiki link actually treats the same equation as you are looking at! Sometimes you're lucky I guess.

Thank you for your reply. I will take a look into the wiki link.