A closed loop system has the characteristic equation given by
$${s^3} + K{s^2} + \left( {K + 2} \right)s + 3 = 0.$$ For this system to be stable, which one of the following conditions should be satisfied?

Let a causal $$LTI$$ system be characterized by the following differential equation, with initial rest condition
$${{{d^2}y} \over {d{t^2}}} + 7{{dy} \over {dt}} + 10y\left( t \right) = 4x\left( t \right) + 5{{dx\left( t \right)} \over {dt}}\,\,$$

Where, $$x(t)$$ and $$y(t)$$ are the input and output respectively. The impulse response of the system is ($$u(t)$$ is the unit step function)

The transfer function of the system $$Y\left( s \right)/U\left( s \right)$$ , whose state-space equations are given below is:
$$\eqalign{ & \left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet \left( t \right)} \cr {\mathop {{x_2}}\limits^ \bullet \left( t \right)} \cr } } \right] = \left[ {\matrix{ 1 & 2 \cr 2 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}\left( t \right)} \cr {{x_2}\left( t \right)} \cr } } \right] + \left[ {\matrix{ 1 \cr 2 \cr } } \right]u\left( t \right) \cr & y\left( t \right) = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}\left( t \right)} \cr {{x_2}\left( t \right)} \cr } } \right] \cr} $$

The Boolean expression $$AB + A\overline C + BC$$ simplifies to