Sequential radioactive decay

In a chain $N_1\to N_2\to N_3$ with distinct decay constants, the Bateman equation expresses each population as a linear combination of the exponential functions $e^{-{\lambda }_{i}t}$. When two consecutive constants coalesce to $\lambda$, the repeated Laplace transform pole produces a term proportional to $t{e}^{-\lambda t}$.

The Bateman equation gives the populations in a sequential radioactive decay chain. For a parent with population $N_1(0)$ and an initially absent daughter, distinct decay constants give $N_2(t)=N_1(0){\lambda }_1(e^{-{\lambda }_{1}t}-e^{-{\lambda }_{2}t})/({\lambda }_2-{\lambda }_1)$; the first population satisfies $N_1^{\prime }=-{\lambda }_1{N}_1$.