### Extra problem to Section 2.5

Uranium in radioactive decay generates radon, whic also decays radiactively (it is a bit more complicated because there are several isotopes of radon). In this simplified model quantity of uran is $y(t)$ is described by $$y'=-\alpha y, \qquad y(0)=y_0 \label{eq-1}$$ and quantity of radon is $z(t)$ is described by $$z'=\beta y-\gamma z, \qquad z(0)=z_0 \label{eq-2}$$ where $y_0,z_0$ are initial quantity, $\alpha,\beta,\gamma$ are positive coefficients.
• (a) Solve (\ref{eq-1}) and then (\ref{eq-2}) (assume that $\gamma\ne \alpha$).
• (b) Plot graphs $y(t)$ and $z(t)$ as $\alpha=\beta=1$, $\gamma=2$.
• (c) Find at what moment $z(t)$ reaches its maximal value, and this maximal value.