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Math Help - Nonlinear DE similar to a Bernoulli equation

  1. #1
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    Nonlinear DE similar to a Bernoulli equation

    Hi all,

    I've got a nonlinear differential equation of the general form

    y' + f(x)y + g(x) = h(x)(y^n)

    to solve.

    For g(x) = 0 this is your bog-standard Bernoulli equation. I've been trying to think of a way to solve it but haven't managed so far.

    Any ideas would be appreciated.

    Many thanks.

    Brad.
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  2. #2
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    Re: Nonlinear DE similar to a Bernoulli equation

    Quote Originally Posted by bradbrad View Post
    Hi all,

    I've got a nonlinear differential equation of the general form

    y' + f(x)y + g(x) = h(x)(y^n)

    to solve.

    For g(x) = 0 this is your bog-standard Bernoulli equation. I've been trying to think of a way to solve it but haven't managed so far.

    Any ideas would be appreciated.

    Many thanks.

    Brad.
    I am not sure if this is correct, but this would be my thinking.

    When we want to solve a second order linear DE, the solution is the sum of the the homogeneous and the non-homogeneous solutions.

    Since we can write this DE as \displaystyle \begin{align*} y' + f(x)\,y = h(x)\,y^n - g(x) \end{align*}, I would expect you could use a similar process here, where the solution would be equal to the sum of the solutions of \displaystyle \begin{align*} y' + f(x)\,y = 0 \end{align*} (homogeneous case), \displaystyle \begin{align*} y' + f(x)\,y = -g(x) \end{align*} (first order linear case) and \displaystyle \begin{align*} y' + f(x)\,y = h(x)\,y^n \end{align*} (Bernoulli case).

    Is there anything wrong with my thinking?
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  3. #3
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    Re: Nonlinear DE similar to a Bernoulli equation

    Hi Prove It !

    In the case of NON-linear ODE, the solution is NOT the sum of the the homogeneous and the non-homogeneous solutions.

    Hi brandbrad !

    As already said, "For g(x) = 0 this is your bog-standard Bernoulli equation".
    More generally, if g(x) exists, the equation can be analytically solved only in some cases.
    Case n=2 : This is a Riccati equation, which method of solving is known ( transformation into a linear second order ODE)
    Case n=3 : This an Abel equation, which can be analytically solved only in a few particular cases. For example see :
    "An Abel ordinary differential equation class generalizing known integrable classes", E.S. Cheb-Terrab 1,2,and A.D. Roche.
    http://www.google.fr/url?sa=t&rct=j&q="an abel ordinary differential equation"%2C0002059v3&source=web&cd=1&cad=rja&ved= 0CCMQFjAA&url=http%3A%2F%2Farxiv.org%2Fpdf%2Fmath% 2F0002059&ei=dO1GUPfQGejZ0QXot4C4AQ&usg=AFQjCNHVF0 JbDhUtCSYorBYWHFfyz6JbfQ
    In case of n>3, analytical solving in not possible in general, except in cases of particular forms of functions f(x), g(x), h(x). In practice, numerical solving is requiered.
    Last edited by JJacquelin; September 4th 2012 at 11:48 PM.
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  4. #4
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    Re: Nonlinear DE similar to a Bernoulli equation

    Hi Prove It and JJ! Many thanks for your input, much appreciated =)

    As it turns out, this is Chini's equation, and as JJ mentions is only generally solvable for particular n.

    I think I'm just going to linearize it and settle for an approximate solution.

    Many thanks guys!
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