# a peculiar ODE

• Dec 31st 2009, 06:02 PM
mmzaj
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

$\displaystyle \int^{\infty}_{-\infty}Adx=\int^{\infty}_{-\infty}Bdx=1$
• Jan 1st 2010, 06:33 AM
chisigma
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

$\displaystyle \chi$ $\displaystyle \sigma$
• Jan 1st 2010, 06:34 PM
mmzaj
Quote:

Originally Posted by chisigma
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 !!)