I was asked to give this function $G'(t) = -G(t)(S+X(t)) + S\cdot H + R(t)/V$ in terms of $G(s)/R(s)$ and $G(s)/X(s)$ by using the Laplace transform. I know $G(0)=A$ and $X(0)=0$, and all the other functions and constants are unknown to me.
When I transform this equation and I neglect the initial conditions, I get $$sG(s) = -G(s)S - \mathcal{L}((G(t)X(t)) + R(s)/V$$
In order to solve this problem, firstly I set $X(t) = 0$ and find $G(s)/R(s) = 1/(V(s+S))$.
But I am struggling to get $G(s)/X(s)$ because I would have to solve the Laplace transform of $G(t)X(t)$. I know I could use the convolution of these to functions, but they are unknown to me, so it doesn't help.
Some people in my class say there is no solution for neither of the two transfer functions, while others don't agree.
Have I made any mistakes in my development? How would you solve this problem?