Some months ago, I solved a problem that required an approximate solution to the non-linear, second order, ordinary differential equation:
x" = x^2, (1)
but I have since forgotten how to approach this, and am a little confused as to which method, or methods, I should use.
I seem to remember writing the equation as:
x" = (x + k)^2, (2)
where k is a very small perturbation from the trajectory.
The aim is to determine whether or not the trajectory, orbit, or path, is stable, and I am also aware that one can even write u as a truncated series expansion, depending on the accuracy of the solution required, as follows:
x = x_0 + kx_1 + (k^2)x_2 + (k^3)x_3 + . . .
and
d(x)/dt = d(x_0 + kx_1 + (k^2)x_2 + (k^3)x_3 + . . .)/dt
where at t = 0, x(0) = a,
and then substitute at least one of these expansions into equation (1).
I have made many attempts to solve this problem recently (including multiplying both sides of equation (1) by the derivative of x with respect to t, and then integrating - which only partially helps, as I only then get half way to the solution – and I have also attempted using matrices, the Jacobian, its associated eigenvalues, and eigenvectors amongst other techniques), but I am really getting nowhere, and I cannot even find a remotely similar solution in any of my books, nor anywhere at all on the Internet.
I also seem to remember that the original question, up to six months ago, possibly required calculation of the zeros, prior to solution (which I think, in this case would be the two repeated values of x = - k, unless this was related to another question), so I wondered if anyone could suggest one or more appropriate methods of solution to adopt – along with the name of equation (1) and any texts that may highlight this particular ODE, along with further details of solution. I also think that it is related to the ODE, known as the equation of path, in Newtonian Mechanics, perturbation theory and non-linear dynamics, as follows:
u” + u = a + bu^2, (3)
but would certainly appreciate any suggestions for equation (1).
Thanks. SWM
Your equation (1) can be solved as follows:
1) multiply both sides of (1) by x' and integrate w.r.t. t. You get:
(x')^2/2-x^3/3=C_1,
where C_1 is the integration constant (which has an obvious meaning of energy for the system under study).
2) Now we should solve the first-order ODE x'=(2(x^3/3-C_1))^(1/2) which is readily solved by separataion of variables and the answer is expressed via elliptic functions.
See also the link below for further details
http://www.wolframalpha.com/input/?i=x''(t)=(x(t))^2
Thank you for that, Artur. Don't know what went wrong in my working earlier this week - as I previously did exactly the same as you have just suggested. Much appreciated for putting me right on that. I just, finally, obtained the same answer as you did in your response. I also referred to my Newtonian Mechanics book regarding the solution to the equation of path, u" + u = a + u^2, where a = constant (particularly regarding whether or not a trajectory, or orbit, is stable), for my own general solution, above, prior to your posting. I will, of course, check this for the usual errors, as I do make them, but I still thank you very much for your valuable contribution. S
If your diff.equation is time dependent ,then the best way to apply He'formalism. or
Laplace transformation.
However if it is time independent you can use series solution method or factorization method .
If a simple approximate solution is enough good for your purposes (instead of the exact one, expressed by elliptic functions), try one of the following approximate analytical methods, easy to implement for your equation: Adomian decomposition method, homotopy perturbation method or variational iteration method.
Use Daftardar-Gejji and Jafari (DJM) to obtain approximate analytical solutions of your problem.
see the following paper is useful to get answer of your problem
As prof. Arthur mentioned. This equation can be solved easily using separation of variables method.
Good luck.
Mason
Use series solution method. If y'' = x^2 then put y=(1/12) x^4
Hope you will use this methodvfor your actual problem.
B.Rath
Dear Fellow Researchers,
Thanks to all of you for your help and suggestions regarding my ODE question.
I now have the correct answer, which is the Weierstrass elliptic function.
Getting to the partial solution, x' = Sqrt[(2x^3 + k)/3] was very easy but, beyond this, it is very complex and involved. Mathematica was a great help over one year ago in helping me to find the complete solution beyond the tips that all of you initially offered me.
I guess that it is also possible to expand the (2x^3 + k)/3)^0.5 part and attempt to solve the original ODE by other means, such as by using regular and singular points - as I think some of you suggested - but I do not need to know how to do any of this at this stage.
At the moment, I am involved more so with PDEs, and any advice upon my current work (or PDE questions and research) will be warmly welcomed.
Kind regards for now,
Stephen Mason
For those who are interested, I have just recorded a YouTube video lecture in which I solve a very similar nonlinear ODE to the one cited in the above question, with the one in my video being 2y'' = 3y^2, subject to the initial conditions, y(0) = 1 and y'(0) = 1. This can be found at: https://youtu.be/ib1EAE32cvo . Hope you both like and enjoy my video, so do let me know.
@Stephen Mason
I suggest better work on
y" =\pm L x^2
which can physical relevance in typical cases .
B.Rath
Biswanath Rath
I am having a bit of trouble interpreting your message. Can you be more specific and write the equation more clearly as I don't know what y" =\pm L x^2 means?
Thanks.
S. Mason
@Stephen Mason
In fact I mean
y"(x)= L x^2
In a simplified language second derivative of
y with respect to x = L x^2
Where L = constant
Thank you Luis,
The method that I normally adopt is to let u = x' = dx/dt, such that, using the Chain Rule, x'' = u' = udu/dx. Then, one obtains udu/dx = x^2, leading to the Integral of udu = Integral of (x^2)dx, thus yielding the solution of u^2 = (2/3)(x^3 + C), where C is a constant of integration. Since we know from above that u = dx/dt, this solution can then be re-written as (dx/dt)^2 = (2/3)(x^3 + C), or Integral of dx/Sqrt[(x^3 + C)] = Sqrt[2/3] times the Integral of dt. I think that the answer from here gets very complicated (due to the Integral of dx/Sqrt[(x^3 + C)]), although having initial conditions, as in my video, helps a lot in obtaining a simpler particular solution.
Kind regards.
Stephen
- Badges
- Science topic
- Similar topics
- Social Psychology
- Attitude