How do we solve the following equation using Green's function?

Question

$$y'' +k^2 y=e^{-\alpha|x|} $$

And I know that the Green's Function is: $$G(x,x') = -\frac{i}{2k} e^{ik|x-x'|} $$

How do I go about finding the solution?

Answer 1

This is a bit more general, but also more insightful.

For every nontrivial polynomial of deg. $n$, one can find a distribution $\phi \in \mathscr{D}'(\mathbb{R}^n)$ such that (see the book of H${\ddot{\text{o}}}$rmander)

$$ p(D)\phi = \delta $$

Then, for every $f \in \mathscr{D}(\mathbb{R}^n)$ it follows that the solution $u$ can be written as $u = \phi * f$, and we see that indeed

$$ p(D)u = p(D)(\phi * f) = \big(p(D)\phi\big)*f = \delta * f = f $$

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In your case, $p(D) \equiv \partial_x^2 + k^2$, $\phi \equiv G$ and $f \equiv e^{-\alpha|x|}$. The solution is then $$ u(x) = G(x,s) * f(s) = \int G(x,s)f(s)\ \mathrm{d}s $$