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A strange ODE

  1. Dec 31, 2009 #1
    what is the solution for y in this peculiar ODE ?

    [tex]A\left(y,x\right)=\frac{dy}{dx}+B(x)(1-y)[/tex]

    with initial conditions :

    [tex]\frac{dy}{dx}=\left0 \ldots , y=0[/tex]

    [tex]\frac{dy}{dx}=\delta(x-x_{0})\ldots,y=1[/tex]

    moreover

    [tex]\int^{\infty}_{-\infty}Adx=\int^{\infty}_{-\infty}Bdx=1[/tex]
     
  2. jcsd
  3. Jan 1, 2010 #2

    berkeman

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    Staff: Mentor

    What is the context of your question? Is this schoolwork?
     
  4. Jan 1, 2010 #3

    HallsofIvy

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    "A(x,y)" implies that A is a function of the independent variables x and y but then dy/dx makes no sense.
     
  5. Jan 1, 2010 #4
    yes and no , it's not a homework , it's for a term paper . the functions A&B are PDFs of some kind , and y is a cumulative distribution function .
     
  6. Jan 1, 2010 #5
    ok , here is a trial :

    [tex]\frac{dy}{dx} = \alpha(x)\cdot y + \beta (x)[/tex] (1)

    where ...

    [tex]\alpha(x)= B(x)[/tex]

    [tex]\beta(x)= A(x) - B(x)[/tex]

    (1) is a linear first order DE and its general solution is...

    [tex]\ y(x)= e^{\int \alpha(x)\cdot dx} (\int \beta(x)\cdot e^{-\int \alpha(x)\cdot dx}\cdot dx + c)[/tex]

    The constant c is derived [if possible...] from the initial conditions
     
  7. Jan 1, 2010 #6
    you can think of [tex]\frac{dy}{dx}[/tex] as an implicit derivative rather than explicit
     
    Last edited: Jan 1, 2010
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