I have started to learn about systems represented by differential equations in Oppenheim's Signals & Systems, and I got really confused about it. I am trying to understand how I can show that a system is LTI (or not), causal (or not) and BIBO stable (or not) for any input.
I know the following (please correct me if I'm wrong):
If the differential equation has zero auxiliary conditions then the system is linear. Therefore, showing linearity is easy.
A system is defined by the response to the Dirac input.
Causal, LTI $\iff$ initial rest (mathematically meaning if $x(t)=0$ for $t<0$ then $y(t)=0$ for $t<0$).
I am not sure if I got it right but if I'm willing to use #3 in order to show that the system is causal and LTI, then it has to have initial rest for any input. How can something like that be shown if as I say it has to be for any $x(t)$? I thought that maybe using #2 will be helpful (solving for the differential equation for delta function and then somehow to show), can it be?
In conclusion, I would thankful if you explain to me how show those features when I got a system that represented by differential equation. For example, in case that we got that simple ODE:
$$\frac{d^2y}{dt^2}-3\frac{dy}{dt}+2y(t)=x(t) \ \ \ ,y(0)=\frac{dy}{dy}=0$$