A system with an input x(t) and output y(t) is described by the r

SOLUTION

To check linearly,

Let, \({y_1}\left( t \right) = t{x_1}\left( t \right)\)

\({y_2}\left( t \right) = t{x_2}\left( t \right)\)

Let, \({y_3}\left( t \right) = {y_1}\left( t \right) + {y_2}\left( t \right) = \underbrace {t\left( {{x_1}\left( t \right) + {x_2}\left( t \right)} \right)}_{\begin{array}{*{20}{c}} \uparrow \\ {{x_3}\left( t \right)\;} \end{array}}\)

Let, \({x_3}\left( t \right) = {x_1}\left( t \right) + {y_2}\left( t \right)\)

\({y_3}\left( t \right) = t{x_3}\left( t \right)\)

Hence, the system is linear

For time variance,

Let input time be shifted to,

\(\Rightarrow y\left( {t - {t_0}} \right) = \left( {t - {t_0}} \right)x\left( {t - {t_0}} \right)\)

If \(x\left( {t - {t_0}} \right)\) the delayed input is given to system

\(y\left( t \right) = tx\left( {t - {t_0}} \right)\)