Chapter 6.5, Problem 22P

The matrix of coefficients for the system of differential equations describing the radioactive decay process in Problem 17, section 6.3, is

$\text{A}=\left(\begin{array}{ccc}-k_1& 0& 0\\ k_1& -k_2& 0\\ 0& k_2& 0\end{array}\right)$.

Use a computer algebra system to find the fundamental matrix $e^{\text{At}}$ and use the result to solve the initial value problem $\text{m'=Am,m}\left(0\right)={\left(m_0,0,0\right)}^T$, where the components of $\text{m}={\left(m_1,m_2,m_3\right)}^T$ are the amounts of each of the three substances $R_1,R_2$ and $R_3$, the first two of which are radioactive, while the third is stable.