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Analytical solution of nonlinear ordinary differential equation

  1. Nov 6, 2013 #1
    Dear All,

    I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative solution.

    [tex]\frac{dx}{dt} = 2Wx + 2xy - 4x^{3}[/tex][tex]\frac{dy}{dt} = \gamma \, (x^{2} - y)[/tex]

    Kindly help me with any methods you that might work and it will be great if you can provide few references where i can read about those methods.

    Also If somebody can help me about how I can use fixed point analytic method to solve this differential equations and some references on it, will be very useful too.

    Thanks a lot in advance.

    PS. I tried homotopy perturbation analysis and simple iteration procedure to try to solve it and it diverges after some time(good only for early short times).
     
    Last edited by a moderator: Nov 14, 2013
  2. jcsd
  3. Nov 14, 2013 #2
  4. Nov 14, 2013 #3
    I would multiply the first equation by x, then reexpress the equations in terms of z, where z = x2. I would also solve for the long time solution, which is z = y = W. How do W and γ compare in magnitude? The answer to this question is important. Also, consider the behavior at long times, less than t = ∞, by writing z = W+εz and y = W+εy. What are the initial conditions on the problem? This could also be relevant.

    Chet
     
  5. Nov 14, 2013 #4
    Thanks for your answer.

    W is between 0 and 1.

    γ is between 0 to ∞.

    Initial conditions are close to zero but cannot be zero.

    It will be very kind of you if you can elaborate on your answer.

    Thanks,
    nitin
     
  6. Nov 14, 2013 #5
    What is the ratio of W to γ in typical situations? This ratio controls everything in the solution trajectory. Have you done what I suggested about replacing x by z? You can also reduce the equations to dimensionless form, at least in terms of the dependent variables, by setting z = Wz* and y = Wy* and solving for z* and y*. This will show that the time constant for the first equation is 1/W, and the time constant for the second equation is 1/γ. Please make the substitutions I have suggested, and show us what you got.

    Chet
     
  7. Dec 5, 2013 #6

    epenguin

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    :shy: I think it is usually worth trying to differentiate such equations and hope to be able to eliminate y and y', then get an equation in only x, x' and x'' which may be solvable analytically. It seems to me you get here a second order linear (not constant coefficients) which you can then solve reducing it to a first-order eq. in variables x and p (= x'), ridding it of second derivative by x'' = p.dp/dx.
    This idea is explained in para. 70 of Piaggio's DE book (switching around notation) but I think I've just said it. It is rather lengthy calculation by my standards and I have not completed a calculation, so am not yet confident.

    I have not yet been able to make out whether this
    is the same method or a better or more general one and its rationale and I too would be grateful for some further elaboration. The quoted text reads like a confusing mistranslation - should 'is' be 'be'?
     
    Last edited: Dec 5, 2013
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