Q$1.$ Use the Laplace Transformation to solve the following linear differential equation system.

Q$2.$ A room contains fresh air initially. At the beginning, the mixture of smoke of cigarette and air is introduced into the room at a rate of $0\mathrm{.}02\frac{m}{m}in$ that involve $0\mathrm{.}1m^{3}C\frac{O}{m^3}$ air. The room is well circulated and mixture is allowed to leave the room at a rate of $0\mathrm{.}02\frac{m^3}{m}in.$

Assume that the densities of gases are equal and $V_{\text{r}\text{o}\text{o}\text{m}}=50m^3.$

a$)$ Derive a mathematical model and use Laplace Transformation method to find the concentration of $CO$ at any time.

b$)$ Calculate the concentration of $CO$ in the room after $2h.$

Q$3.$ Consider the following boundary and initial conditions and solve the given partial differential equation using the separation of variables method

for $t>0$

for $t>0$

for $0x4$

for $0x4$

Q$4.$ An infinitly long string having one end at $x=0$ is initially rest on the $x$ axis. The end $x=0$ undergoes a periodic transverse displacement given by $u=t.$ Find the displacement of any point on the string at any time using laplace transformation method if $=1.($find the solution in $x,t$ domain$)$