Not sure if this is the correct sub stack, but here's my problem
In a given LTI System, of which the transfer behaviour is described by differential equation.
$$\sum_{i=0}^{{n}}A_{i}\frac{d^{i}\alpha}{dx^{i}} = \sum_{i=0}^{{n}}B_{i}\frac{d^{i}\beta}{dx^{i}}$$ With Initial condition $$\frac{d^{i}\alpha}{dt^{i}}(x_{0})=\alpha_{0}^{^{(i)}} , i = 0,1,2,...,n-1$$
What's the requirements on the coefficient $A_{i}$ and $B_{i}$ in the equation for the system to be stable?
This differential equation can also be translated to the following transfer function,
$$ G(s) = \frac{\sum_{k=0}^n B_k \, s^k}{\sum_{k=0}^n A_k \, s^k}, $$
which is only stable if the roots of the polynomial in the denominator are located in the left half plane, these roots are also called the poles of $G(s)$. The values of $B_k$ do not matter, unless some roots of the polynomial of the numerator, also called the zeros of $G(s)$, are also poles, such that you get pole-zero cancellation.