How can I find the transfer function of the following block diagram? I've the following image and I want to find the transfer function from input $x(t)$ to output $y(t)$.
I know that I have to apply Laplace Transform, so the integrator becomes $\dfrac{1}{s}$, but I don't know what to do with the numbers $a$ and $b$. Should it still be the same?
If a and b still the same, then I found that $H(s) = \dfrac{Y(s)}{X(s)} = s^2-as-b$.
In such cases it can be useful to introduce auxiliary variables at the input of the integrators. For the given diagram you could use a signal $u(t)$ at the input of the second integrator. The equation for its Laplace transform $U(s)$ becomes
$$U(s)=\frac{1}{s}\big[X(s)-bY(s)\big]-aY(s)\tag{1}$$
You need another equation relating $U(s)$ to $Y(s)$, but that one is trivial. From those two equations you can express $Y(s)$ in terms of $X(s)$, and in this way you can obtain the transfer function $H(s)=Y(s)/X(s)$. The solution in your question is not correct. The correct solution must be a rational function, not a polynomial.