Obtain a state variable representation of the system governed by the differential equation: $${{{d^2}y} \over {d{t^2}}} + {{dy} \over {dt}} - 2y = u\left( t \right){e^{ - t}},\,\,\,$$ with the choice of state variables as $${x_1} = y,$$ $${x_2} = \left( {{{dy} \over {dt}} - y} \right){e^t}.$$ Aso find $${x_2}\left( t \right),$$ given that $$u(t)$$ is a unit step function and $${x_2}\left( 0 \right) = 0.$$

For the circuit shown in Fig. the Boolean expression for the output $$Y$$ in terms of inputs $$P,$$ $$Q,$$ $$R$$ and $$S$$ is

The frequency of the clock signal applied to the rising edge triggered $$D$$ flip-flop shown in Fig. is $$10$$ $$kHz.$$ The frequency of the signal available at $$Q$$ is

The ripple counter shown in Fig. is made up negative edge triggered $$J$$-$$E$$ flips flops. The signal levels at $$J$$ and $$K$$ inputs of all the flip-flops are maintained at logic $$1$$. Assume that all outputs are cleared just prior to applying the clock signal.

$$(a)$$ Create a table of $${Q_0},{Q_1},{Q_2}$$ and $$A$$ in the format given below for $$10$$ successive
$$\,\,\,\,\,\,\,\,$$ input cycles of the clock $$CLK1.$$
$$(b)$$ Determine the module number of the counter.
$$(c)$$ Modify the circuit of Fig. to create a modulo$$-6$$ counter using the same
$$\,\,\,\,\,\,\,$$ components used in the figure.