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Math Help - Reducing non-linear ODE to Linear

  1. #1
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    Reducing non-linear ODE to Linear

    Hi all,

    I'm struggling with the following problem.

    Consider basic non linear differential equation:

    dx/dt + a(t)x = b(t)xn

    I understand that I can define a new variable y = x1-n

    However I am unsure how to make the general substituion to get to the solution:

    dy/dt + (1-n)a(t)y = (1-n)b(t)

    I would really appreciate it if someone could point me in the right direction and provide me with the steps.

    I have tried differentiating it and substituting in for x dot but I still have x terms floating around.

    Thank you for the help!
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  2. #2
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    Re: Reducing non-linear ODE to Linear

    Differentiate y=x^{1-n} with respect to t so that \frac{dy}{dt}=\frac{1-n}{x^n}\frac{dx}{dt} or \frac{dx}{dt}=\frac{x^n}{1-n}\frac{dy}{dt}.

    Sub it in along with the other bit then multiply through by 1-n and divide by x^n.
    Last edited by a tutor; February 13th 2013 at 09:20 AM.
    Thanks from acid_rainbow
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  3. #3
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    Re: Reducing non-linear ODE to Linear

    Thank's it just clicked. Much appreciated
    Last edited by acid_rainbow; February 13th 2013 at 09:53 AM.
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