Chapter 1.1, Problem 6E

In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law:
The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used.
6. Model radioactive decay using the notation
  t = time (independent variable),
  r(t) = amount of particular radioactive isotope
  present at time t (dependent variable),
  − $\lambda$ = decay rate (parameter).
Note that the minus sign is used so that $\lambda$ > 0.
(a) Using this notation, write a model for the decay of a particular radioactive isotope.
(b) If the amount of the isotope present at t = 0 is r₀, state the corresponding initial-value problem for the model in part (a).