$6\mathrm{.}3.$ The system shown in Fig. P$6\mathrm{.}1($a$)$ is subjected to a step input $x(t)=a.$

suddenly applied at $t=0$ and lasting for time $T.$ At $t=T,x(t)$ suddenly

returns to zero. Assuming that $x(t)=0$ for $t<mn>0</mn>$ and $t>T,$ determine

the system's response $y(t)$ to this input:

$($i$)$ by classical methods of solving the governing differential equation

of motion;

$($ii$)$ by applying the convolution integral $(6\mathrm{.}24)$:

and

$($iii$)$ by determining the Fourier transform of the input, applying $(6\mathrm{.}20),$

and then calculating the inverse transform of $Y().$