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Suppose I have [tex]\epsilon^2 y''' - y' = \frac{1}{1+x^2}[/tex].

The goal is to calculate the fourier transform of [tex]y(x,t)[/tex] where we define,

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

We're also given that,

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

Now we take transforms of both sides:

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

[tex]\rightarrow -i k^3 \epsilon^2 \hat{y} - ik \hat{y} = \pi e^{|k|}[/tex]

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

The answer, however, is supposed to be:

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

where 'a' is some constant.

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

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# Where did this term come from? (Fourier)

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