Example $1.($Only the First Problem in the Review for the Midterm Exam$)$

A radioactive material has a decay rate proportional to the amount of radioactive material present at that time, with a proportionality factor of $2$ per unit time.

$($a$)$ Write a differential equation of the form $\backslash ($ P$^\{\backslash$prime$\}=$F$($P$)\backslash ),$ which models this situation, where P is the amount of radioactive material $($measured in micrograms$)$ as a function of time.

$($b$)$ Now, assume that we have the same radioactive material decaying as above, but we are adding additional material $($of the same type$)$ at a constant rate of $6$ micrograms per unit time. Write the differential equation in this case.

$($c$)$ Solve the ordinary differential equation in part $($b$)$ above, assuming the initial amount of radioactive material is $70$ micrograms.