This is the long$-$term behaviour of the system.

Example $3\mathrm{.}6$: Initial and final value theorems

Suppose a process can be described by the following linear differential equation:

$\frac{dy}{dt}+y=Ku(t)$

in which $y$ is the process output, $u$ the process input, $$ the process time constant and $K$ the process gain.

The initial conditions are as follows: $y(0)=0$

If $u(t)=A$ :

$($a$)$ Determine the Laplace transform of the differential equation.

$($b$)$ Determine the long$-$term and short$-$lerm behaviour of this system.