Determine the system function H(s) of a system and find out the differential equation I have created the following system for practice purposes. From this system I want to determine the system function H(s).



In the picture I have worked with auxiliary (dummy) variables, which should simplify the whole calculation (see blue marking).

How did I approach the matter now? Well, first I have drawn in my auxiliary variables, then I have looked at the summing elements.

I know the relation $H(s)=\frac{Y(s)}{X(s)}$, so I have to transfer my equations into this form.

So these are my equations:

$$\text{Eq 1:}\quad X(s) + G(s) = P(s)$$
$$\text{Eq 2:}\quad 3P(s)=Y(s),$$
$$\text{Eq 3:}\quad G(s) = s^{-1}Y(s)+2Y(s)$$ 
Eq 3 in 1 results in:
$$X(s) + s^{-1}Y(s)+2Y(s) = P(s)$$
Using Eq. 2 results in:
$$Y(s) = 3X(s) + 3s^{-1}Y(s) + 6Y(s)$$
Therefor:
$$Y(s)(-5-3s^{-1})= 3X(s)$$
$$H(s)=\frac{Y(s)}{X(s)}=\frac{3}{-5-3s^{-1}}$$

My first question is, if you see the system function $H(s)$, is it correct so far?

Then I would be interested in the conversion into a differential equation, here I have already prepared something:

$$Y(s)(-5-3s^{-1})= 3X(s)$$

This should now be handled with the inverse Laplace transformation:

$$-5y(t)+\mathcal{L}^{-1}(-3s^{-1})= 3x(t) \\-5y(t)-3\int_{0}^{\tau} x(\tau) d\tau = 3x(t)$$

My problem at this point is that when I look into my Laplace transform table, I find that $s^{-1}Y(s)$ transformed back results in an integral. Now it is said that these systems can also be given in a differential equation (in terms ot the z-trafo this would be comparable to the difference equation there). How would one come to this differential equation? I have a hunch but I would be interested in your answers :)

I hope my question is so far understandable. Thank you very much.
Your transfer function looks correct. Note that you can rewrite $H(s)$ as

$$H(s)=-\frac{3s}{5s+3}\tag{1}$$

or, equivalently,

$$5sY(s)+3Y(s)=-3sX(s)\tag{2}$$

Since multiplication by $s$ corresponds to differentiation in the time domain you obtain from $(2)$

$$5y'(t)+3y(t)=-3x'(t)\tag{3}$$