How to conclude LTI, causality and BIBO stability of a system represented by a differential equation?
I take your equation:
$\frac{d^2y}{dt^2}-3\frac{dy}{dt}+2y(t)=x(t)$

Laplace transform will be:
$ s^2Y(S) - 3sY(S) +2Y(s) = X(s) $

Now I can find transfer function: $H(s) \triangleq \frac{Y(s)}{X(s)}$

$H(s)=\frac{1}{s^2-3s+2}$

Poles are the solution of denominator = 0 and zeros are the solution of numerator = 0. In this case there are not zeros because numerator is always equal to 1, denominator = 0 is a second order equation that can be easily solved.

$ Poles[H(s)] = \{s=1 ; s=2\} \\ Zeros[H(s)] = \{\} $

There are two real poles.

To determine if the system is BIBO stable you can see if all poles have real part $ &lt; 0 $. In this is example there is at least one pole that has real part  $ \geq 0 $ (both in reality) so this system is UNSTALBE.

To be causal the number of zeros must be $\leq$ of the number of poles in this case we have $ 0 \leq 2 $ that it's true and means that your system is CAUSAL.

Time invariant system is a system that doesn't depends on absolute time but only on relative time from time origin. In a transfer function generally there are not information to determine if it's a time invariant system or not beacuse it should be an external specification of your problem. Btw in DSP you consider systems time invariant almost ever. 

Finally this is only a quick overview of zero-pole analysis you should study the argument as well and the prove of the concepts I  used.