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Thread: a peculiar ODE

  1. #1
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    a peculiar ODE

    what is the solution for y in this peculiar ODE ?

    $\displaystyle A\left(y,x\right)=\frac{dy}{dx}+B(x)(1-y)$

    with initial conditions :

    $\displaystyle \frac{dy}{dx}= 0 \ldots,y=0$

    $\displaystyle \frac{dy}{dx}=\delta(x-x_{0})\ldots,y=1$

    moreover

    $\displaystyle \int^{\infty}_{-\infty}Adx=\int^{\infty}_{-\infty}Bdx=1$
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  2. #2
    MHF Contributor chisigma's Avatar
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    The condition...

    $\displaystyle \int_{-\infty}^{+\infty} A(x,y)\cdot dx=1$ (1)

    ... means that $\displaystyle A(*,*)$ is a function of the only $\displaystyle x$ so that the DE is...

    $\displaystyle \frac{dy}{dx} = \alpha(x)\cdot y + \beta (x)$ (2)

    ... where ...

    $\displaystyle \alpha(x)= B(x)$

    $\displaystyle \beta(x)= A(x) - B(x)$ (3)

    The (2) is a linear first order DE and its general solution is...

    $\displaystyle y(x)= e^{\int \alpha(x)\cdot dx} \{\int \beta(x)\cdot e^{-\int \alpha(x)\cdot dx}\cdot dx + c\}$ (4)

    The constant c is derived [when possible...] from the 'initial conditions' ...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Last edited by chisigma; Jan 1st 2010 at 06:48 AM.
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  3. #3
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    Quote Originally Posted by chisigma View Post
    The condition...

    $\displaystyle \int_{-\infty}^{+\infty} A(x,y)\cdot dx=1$ (1)

    ... means that $\displaystyle A(*,*)$ is a function of the only $\displaystyle x$ so that the DE is...

    $\displaystyle \frac{dy}{dx} = \alpha(x)\cdot y + \beta (x)$ (2)

    ... where ...

    $\displaystyle \alpha(x)= B(x)$

    $\displaystyle \beta(x)= A(x) - B(x)$ (3)

    The (2) is a linear first order DE and its general solution is...

    $\displaystyle y(x)= e^{\int \alpha(x)\cdot dx} \{\int \beta(x)\cdot e^{-\int \alpha(x)\cdot dx}\cdot dx + c\}$ (4)

    The constant c is derived [when possible...] from the 'initial conditions' ...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    thanks , that was really helpful . but i stated the simplest form the problem in hand , where A , B , and y are functions of the single variable x . but the real case requires that the latter functions could be functions of an arbitrary number of independent variables $\displaystyle x_{1},x_{2}..x_{n}$ and the equation becomes :

    $\displaystyle A(x_{1},x_{2}..x_{n})=\frac{dy}{d\Omega}+B(x_{1},x _{2}..x_{n})(1-y)$
    where $\displaystyle d\Omega=\prod^{n}_{i=1} dx_{i}$
    is a unit volume (so to speak !!)

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