Green's Function for a Differential Equation

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We consider the equation

,

that should be solved for x (- , + ). On applying Fourier method we multiply the whole equation by e - i  x and integrate it over (- , + ) by x :

We suppose that Green function and its first derivatives should vanish at infinity and make integration by parts in the first term twice to get

The integrals in the left hand side are the Fourier images of G that we denote as . On applying the  - function property we calculate the right hand side to get the algebraic equation

(  - 2 ) =

Thus, we have = Taking the inverse Fourier transform we get

= ...