Chapter 5.4, Problem 26P

A radioactive substance $R_1$ having decay rate $k_1$ distintegrates into a second radioactive substance $R_2$ having decay constant $k_2\ne k_1$. Substance $R_2$ disintegrates into $R_3$ which is stable. If $m_i\left(t\right)$ represents the mass of substance $R_i$ at time $t$, $i=1,2,3$, the applicable equations are

$\begin{array}{l}{m}_1\text{'}=-k_1{m}_1\\ m_2\text{'}=k_1{m}_1-k_2{m}_2\\ m_3\text{'}=k_2{m}_2\end{array}$

Use the Laplace transform to solve this system under the conditions

$m_1\left(0\right)=m_0,m_2\left(0\right)=0,m_3\left(0\right)=0$.