Two Stage Radioactive Decay

Theorem
Let $R_A$ be a radioactive isotope which decays into another radioactive isotope $R_B$ with the rate constant $k_1$.

Let $R_B$ decay into a third element $R_C$ with the rate constant $k_2$.

Let the initial quantity of $R_A$ be $x_0$.

Let the amounts of $R_A$ and $R_B$ present at time $t$ be $x$ and $y$ respectively.

Then:
 * $y = \begin{cases}

\dfrac {k_1 x_0} {k_2 - k_1} \left({e^{-k_1 t} - e^{-k_2 t} }\right) & : k_1 \ne k_2 \\ & \\ k_1 x_0 t e^{-k_1 t} & : k_1 = k_2 \end{cases}$