I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform then we will have $$a_0sY(s) + a_1Y(s)=b_0X(s)+b_1/s$$ The problem is that i have to reach $X(s)/Y(s)$ .
What is the Laplace transform transfer function of affine expression $\dot x = bu + c$? Based on this we do not have a solution?
What you could do is to introduce a new variable $\bar{x}(t) := x(t) + \frac{b_1}{b_0}$. This would enable you to express the transfer function by means of $\bar{X}(s)$, at least:
$$ \frac{\bar{X}(s)}{Y(s)} = \frac{a_0s+a_1}{b_0}. $$