I am trying to find the inverse system of the following (I tried finding the mathematical inverse function but since it is not the same I am not so sure) . Can someone help me find it?
$$ y(t)=\int_{t-1}^{t+1}\cos\left(\frac{\pi\tau}{8}\right)x(\tau)d\tau $$
It seems that we have the same homework. You probably are Greek.
After having the same question, I came to the conclusion that in order for this system to be invertible, you have to prove that for any given $$x_1(t), \, x_2(t)$$ the following sentence is correct: $$ x_1(t) \ne x_2(t) \,\, {\Rightarrow} \,\, y_1(t) \ne y_2(t) $$
As like you proved when a math function is $"1-1"$.
For this very example, if you have the following the following two signals:
$$ \ x_1(t) = δ(t-4) \ $$
and
$$ x_2(t)=0 $$
it turns out that while initially $ x_1(t) \ne x_2(t) $, we have $ cos(πt/8)x_1(t) = cos(πt/8)x_2(t) $, for any $t$. This means that
$$ \int_{t-1}^{t+1}\cos\left(\frac{\pi\tau}{8}\right)x_1(\tau)d\tau = \int_{t-1}^{t+1}\cos\left(\frac{\pi\tau}{8}\right)x_2(\tau)d\tau $$
and the system is not invertible.
HINT:
The given input-output relation can be interpreted as modulation followed by filtering with an LTI system:
$$\tilde{x}(t)=x(t)\cos(\pi t/8)\\y(t)=\int_{t-1}^{t+1}\tilde{x}(\tau)d\tau$$
First you need to figure out the impulse response of the LTI system. Then find its frequency response. Next, see if that frequency response can be inverted or not. The modulation can be undone by appropriate demodulation. So the main question to solve is if the LTI system can be inverted. You can see this by checking if you can divide by its frequency response.
