Where did this term come from? (Fourier)

[rsq_a]

A simple question.

Suppose I have $$\epsilon^2 y''' - y' = \frac{1}{1+x^2}$$.

The goal is to calculate the Fourier transform of $$y(x,t)$$ where we define,

$$\hat{y}(k,t) = F[\phi] = \int_{-\infty}^{\infty} y e^{ikx} ds$$

We're also given that,

$$F\left[ \frac{1}{1+x^2} \right] = \pi e^{-|k|}$$

Now we take transforms of both sides:

$$\rightarrow F[\epsilon^2 y''' - y] = F[\frac{1}{1+x^2}]$$

$$\rightarrow -i k^3 \epsilon^2 \hat{y} - ik \hat{y} = \pi e^{|k|}$$

$$\rightarrow \hat{y} = -\frac{\pi e^{-|k|}}{ik(1-\epsilon^2 k^2)}$$

The answer, however, is supposed to be:

$$\hat{y} = -\frac{\pi e^{-|k|}}{ik(1-\epsilon^2 k)} + 2\pi a \frac{k\delta(k)}{k(1-\epsilon^2 k^2)}$$

where 'a' is some constant.

My question is why? I know it has something to do with an additive constant, but I need someone to be explicit with the mistake.

 

[rsq_a]

I've figured out a way to 'see' the result:

Instead write the equation like this:

$$\epsilon^2 y''' - y' = \frac{1}{1+x^2} + \frac{d}{dx} 2\pi{a}$$

and the result follows automatically.

But what I don't understand is why I 'need' to do this. Why haven't I seen this in any texts on Fourier transforms (that if one is applying the transform to a (for example), third order equation reducible to second order, one needs to add arbitrary constants.