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**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

$\displaystyle \chi$ $\displaystyle \sigma$