Exercises

1. Radioactive isotopes decay according to a linear first order ODE that produces an exponential solution. If the daughter product of the decay process also undergoes radioactive decay, then its population is determined by a second linear first order ODE that is coupled to the first ODE as shown below. Note that the decay rate $N_1(t)$ that reduces the number of $N_1$ serves as a positive contribution to the production of the daughter isotope $N_2(t)$ in the second equation. In this way, decay chains of radioactive isotopes are mathematically modeled as coupled linear first order ODEs.

   $$\begin{array}{l}\frac{d{N}_1(t)}{dt}=-{\lambda }_1{N}_1(t)\\ \frac{d{N}_2(t)}{dt}={\lambda }_1{N}_1(t)-{\lambda }_2{N}_2(t)\end{array}$$
   Use the initial conditions: $N_1(t)$ at t=0 is 1000 and $N_2(t)$ at t=0 is 0.