I am having a really hard time trying to apply the Fourier transform to a second order ODE. The questions ask to find a solution to the ODE with the following conditions,
$$\frac{d^2y}{dx^2} -k^2y = f(x)$$ $$\lim_{|x|\to\infty}y(x)= 0$$ $$\lim_{|x|\to\infty}\frac{dy}{dx} = 0$$
Where k is an arbitrary real constant and I have to solve for the cases, $f(x) = 1$ and $f(x) = \delta(x-x_0)$
This is for a PDE class that I am taking and we are using the Haberman book, So the transforms I am doing are based off of the tables provided in the text.
For the case $f(x) = 1$, I get the following transform,
$$Y(\omega) = -\frac{\delta(\omega)}{(\omega^2 + k^2)}$$
However, looking at the table provided, I am having a hard time trying to find the inverse transform.
I can get something that looks transformable by doing the following,
$$Y(\omega) = -\delta(\omega)\frac{1}{2k}\frac{2k}{(w^2+k^2)}$$
Using convolution leaves me with an answer of,
$$y(x) = \frac{-1}{2k}e^{-k|x|}$$
but I am unsure of this is a correct solution. Any help and further explanation would be appreciated.
Long story short: For $f(x)=1$, the ODE does simply not have a solution (with the given decay conditions).
Disclaimer: I am a little bit confused which version of the Fourier transform you are using since the constants are not consistent, I am guessing the unitary angular freuency version? (https://en.wikipedia.org/wiki/Fourier_transform#Tables_of_important_Fourier_transforms)
Following your idea with the convolution (you made a small error here) yields $$ f(x)=1\Rightarrow Y(\omega)=-\frac{\sqrt{2\pi}\delta(\omega)}{\omega^2+k^2}=-\frac{1}{2k}\cdot\sqrt{2\pi}\cdot\sqrt{2\pi}\delta(\omega)\cdot\sqrt{\frac{2}{\pi}}\frac{k}{\omega^2+k^2}\\ \Rightarrow y(x)=-\frac{1}{2k}\left(1*e^{-k|x|}\right)=-\frac{1}{2k}\int_\mathbb{R}e^{-k|x|}dx=-\frac{1}{k^2} $$ which is a solution of the differential equation by itself but does obviously not satisfy the first decay condition. (Since the ODE is linear and autonomous a more general class of solutions can be found easily which reads $f(x) = c_1 e^{k x} + c_2 e^{-k x} - \frac{1}{k^2}$, however the constants cannot be chosen in a way to fulfill the given conditions.)
The second case $f(x)=\delta(x-x_0)$ however does allow for a solution (following the same procedure) namely $$y(x)=-\frac{e^{-k|x-x_0|}}{2k}$$.