$(3.)(15$ points$)$ A tank with a capacity of $500$ gal initially contains $200$ gal of water with $100$ lb salt mixed in$.$ Water containing $1$ lb of salt$/$gallon is entering at a rate of $3ga\frac{l}{m}in.$ The water is well mixed in the tank. There is a drain that allows the salty water to flow out of the tank at a rate of $2ga\frac{l}{m}in.$

$($a$)$ If $x(t)$ is the amount of salt in the tank as a funciton of time, set up an initial value problem that describes this flow process.

$($b$)$ Find the amount of salt in the tank as a function of time.

$($c$)$ What do you expect to happen after a long time?