P384_Assignment_8_SOLUTIONS - 1 Physics 384 2016 Assignment 8 SOLUTIONS Problem I Consider the inhomogeneous ODE d2 y(x k 2 y(x = f(x dx2 where the

P384_Assignment_8_SOLUTIONS - 1 Physics 384 2016 Assignment...

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1 Physics 384 2016 Assignment 8 SOLUTIONS Problem IConsider the inhomogeneous ODEd2y(x)dx2-k2y(x) =f(x)where the boundary conditions are thaty(±∞) =finite. Use the Fourier transform methodto show that the Green’s functionG(x;x0)for this problem, which satisfiesd2G(x;x0)dx2-k2G(x;x0) =δ(x-x0)(1)can be expressed asG(x;x0) =-12πˆ-∞eik0(x-x0)k02+k2dk0Evaluate the integral and show thatG(x;x0) =-12ke-k|x-x0|x
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