$5.(20$ points$)$ The rate at which a chemical is depleted from a solution is proportional to the concentration of the chemical in the solution.

$($a$)$ Determine the difference equation $($not differential equation$)$ that represents the concentration of the chemical $\backslash ($ C$($t$)\backslash )$ as a function of time.

$($b$)$ Derive the differential equation that represents the concentration of the chemical $\backslash ($ C$($t$)\backslash )$ as a function of time.

$($c$)$ Solve the differential equation with $\backslash ($ C$(0)=$C$\_\{0\}\backslash )$ to obtain $\backslash ($ C$($t$)\backslash ).$