Solving $\frac{d^2u}{dx^2} + 2\frac{du}{dx} + u = e^{-|x|}$ via Fourier Transform

Question

I'm trying to solve the following differential equation: $$\frac{d^2u}{dx^2} + 2\frac{du}{dx} + u = e^{-|x|}$$ subject to $u(\pm \infty) \rightarrow 0$

I use the following Fourier Transform: $\frac{1}{\sqrt{2 \pi}}\int^\infty_{-\infty}f(x)e^{-ikx}$

Using the properties of the Fourier Transform on spacial derivatives and the transform of $e^{-|x|}$, I arrived at the following: $$\tilde{u} = -\sqrt{\frac{2}{\pi}}(\frac{1}{1+k^2})\frac{1}{(k+i)^2}$$

I don't really know how to advance with finding the inverse fourier transform, I thought about using the convolution theorem or maybe the standard formula and then use some contour integration. How should I go about it?

Many thanks!

Answer 1

First, note that

$$-\sqrt{\frac2\pi}\frac1{1+k^2}\frac1{(k+i)^2}=-\sqrt{\frac2\pi}\frac1{(k+i)(k-i)}\frac1{(k+i)^2}=-\sqrt{\frac2\pi}\frac1{k-i}\frac1{(k+i)^3}=-\sqrt{\frac2\pi}\left(\frac1{8i^3(x-i)}-\frac1{2i(x+i)^3}-\frac1{4i^2(x+i)^2}-\frac1{8i^3(x+i)}\right)$$ (see also partial fraction decomposition on wikipedia and on wolframalpha site)

Then, either use a table of known Fourier inverses, or integration by parts, to calculate the inverse fourier transform of each term separately.