The Laplace transform of the casual periodic square wave of period T shown in the figure below is

Let x(t) = a s(t) +s(-t) with s(t) = $$\beta {e^{ - 4t}}u\left( t \right)$$, where u(t) is unit step function. If the bilateral Laplace transform of x(t) is $$$X\left( S \right)\, = {{16} \over {{S^2} - 16}} - 4 < {\mathop{\rm Re}\nolimits} \left\{ s \right\} < 4;$$$

Then the value of β is ______.

The solution of the differential equation $${{h\left( {t + 1} \right)} \over {h\left( t \right)}}\,\,\,\,\,{{{d^2}y} \over {d{t^{ \to 2}}}} + {{2\,dy} \over {dt}} + y\, = \,0$$ with $$\,y\left( 0 \right)\, = \,y'\left( 0 \right)\, = \,1$$ is

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is output of the system $$\mathop y\limits^ \bullet \left( t \right) + 5y\left( t \right) = u\left( t \right)$$

When y(0) = 1 and u(t) is a unit step function, y(t) is