$2-(20$ points$)$ A thermal system consisting of a solid object $($with mass $m=2kg)$ is initially at a uniform temperature of $T_0=20C.$ At $t=0,$ a heat source with a constant heat input $Q=10W$ is applied to the object, causing the temperature to increase. The temperature change follows the equation:

$m\frac{dT(t)}{dt}=-k(T(t)-T_{})+Q$

where:

$T(t)$ is the temperature of the object at time $t,$

$T=50C$ is the ambient temperature,

$k={0\mathrm{.}01}^{**}[$your last $2$ digits of your student ID number$](\frac{W}{}C)$ is the heat transfer coefficient, and

$Q=10W$ is the heat source.

a$)$ Apply Laplace transform and the inverse of Laplace transform to obtain T$($t$).$

b$)$ Discuss the behavior of the temperature as $t$ and determine the steady$-$state temperature.