A tank contains $1000$ litre of water in which $25kg$ of salt is dissolved. Then, $50$ litres of brine, containing $(1+e^{-0\mathrm{.}1t}cost)kg$ of dissolved salt, starts to run into the tank per minute. The mixture, kept uniform by stirring, runs out at the same rate.

a$)$ Find the amount of salt $y(t)$ in the tank at any time $t.$

$($Answer: $($:$y(t)==1+\frac{e^{-0\mathrm{.}1t}}{1002\mathrm{.}5}[(-0\mathrm{.}05cost+sint)]-0\mathrm{.}975e^{-0\mathrm{.}05t}$