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I want to solve:

$$y'(t)+y(t)+y(t-1)=\delta (t)$$

The delta on the right hand side is used instead of the initial condition of $y(0)=1$

I then take the fourier transform of both sides:

$$Y(i w+1+e^{- iw})=1$$

Taking the inverse fourier transform of this expression is hopeless, I attempt to do it numerically. This is what $Y$ looks like (blue is real, red is imaginary) :

enter image description here

This is its numerical inverse transform:

enter image description here

(Don't worry about the oscillation near the discontinuities) But this clearly is not correct since the solution is supposed to look like this:

enter image description here

I would like to learn what I have done wrong, how to do it correctly, and which problem I have actually solved (what is the function in the second image a solution to)

EDIT: I have used an optimization algorithm to estimate the function that represents the fourier transform of the actual solution and it looks like this:

$$\frac{0.431}{i2.704w+0.432+0.430e^{-i6.291w}}$$

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The mistake made in the solution was mixing up cyclic frequency and angular frequency conventions of fourier transform. As can be seen from the expression:

$$\frac{0.431}{i2.704w+0.432+0.430e^{-i6.291w}}=\frac{1}{i2\pi w+1+e^{-i2\pi w}}$$

The $w$ was treated as cyclic frequency, which caused the discrepancy.

In other words, if the numerical inverse transform is taken correctly, the solution will be the same (apart from the ringing artifacts)

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