A $600-$gal large tank initially contains $50$ gallons of brine in which there is dissolved $10$ lb of salt. Brine containing $2l\frac{b}{g}al$ of salt dissolved in water flows into the tank at the rate of $5ga\frac{l}{m}in.$ The water in the tank is kept well mixed, and a drain is opened at the bottom of the tank allowing the solution to leave tank at the rate of $3$ gallons per minute.

a$)$ Derive a $DE$ for the total amount $x(t)$ of salt in the tank at time $t.$