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Math Help - differential equation

  1. #1
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    differential equation

    Hello:

    I have to solve the following differential equation as shown in the attachment

    please help me how to solve this differential equation
    Attached Thumbnails Attached Thumbnails differential equation-diff-eq.jpg  
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  2. #2
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    Re: differential equation

    Hey matmman.

    Have you considered the Bernoulli solution?

    Bernoulli differential equation - Wikipedia, the free encyclopedia
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  3. #3
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    Re: differential equation

    Quote Originally Posted by chiro View Post
    Hey matmman.

    Have you considered the Bernoulli solution?

    Bernoulli differential equation - Wikipedia, the free encyclopedia
    It's not Bernoulli because of the extra \displaystyle \begin{align*} e^{-x} \end{align*} being subtracted...
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    Re: differential equation

    Hi !
    it is a Riccati ODE.
    First, change the variable : t=exp(-x)
    leading to dy/dt = -(y/4)+(y/t)+1
    Then let y= (1/4) (df/dt)(1/f) where f(t) is the new unknown function.
    This leads to a linear second order ODE of the Bessel Kind.
    Solve it for f(t) in termes of t*BeselI[1, t/2] and t*BesselK[1, t/2]
    Then derivate it and bring f(t) and (df/dt) into y= 4(df/dt)*(1/f)
    Finally, remplace t by exp(-x)
    Last edited by JJacquelin; November 4th 2012 at 01:36 AM.
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  5. #5
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    Re: differential equation

    Wolfram gets this...
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    Re: differential equation

    Wolfram gives the result that you will obtain thanks to the method which was given in my preceeding post.
    Note that t=sqrt(exp(-2x)). In the WolframAlpha formula, the denominator of the fraction is the function f(t) and the numerator is 4(df/dt).
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    Re: differential equation

    thanks for all

    can we solve this equation numerically by matlab using ode or bvp4c function
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    Re: differential equation

    hi JJacquelin:
    if we subsitute y by (1/4) (df/dt)(1/f) then how the function will be
    what about dy/dt replace it by (d2f/dt2)(1/4f) and y^2 replace it by (df/dt)^2*(1/4)^2
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    Re: differential equation

    Sorry, there was a typo :
    y= 4 (df/dt)(1/f)
    This was correctly written at the end of my post :
    << Then derivate it and bring f(t) and (df/dt) into y= 4(df/dt)*(1/f) >>
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  10. #10
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    Re: differential equation

    please can you send send the solution in detial
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  11. #11
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    Re: differential equation

    Quote Originally Posted by matmman View Post
    please can you send send the solution in detial
    I will not write the solution in whole details. When we reach the point where the Bessel functions appear, I suppose that the properties of these functions are known. Then, going into details would be a waste of time.
    Attached Thumbnails Attached Thumbnails differential equation-riccati-bessel.jpg  
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