Chapter 2.P2, Problem 2P

Additional processes due to biotic and abiotic degradation contributing to source decay can be accounted for by adding a decay term to $\left(3\right)$ that is proportional to $m\left(t\right)$,

$m\text{'}\left(t\right)=-\alpha m^{\gamma }-\lambda m$, $\left(\text{i}\right)$

where $\lambda$ is the associated decay rate constant. Find solutions of Eq. $\left(\text{i}\right)$ using the initial condition $m\left(0\right)=m_0$ for the following cases: (i) $\gamma =1$, (ii) $\gamma \ne 1$ and $\lambda \ne 0$. Then find expressions for $c_s\left(t\right)$ using Eq. $\left(2\right)$.

Hint: Eq. (i) is a type of nonlinear equation known as a Bernoulli equation. A method for solving Bernoulli equations is discussed in Section $2.7$.

Equation $\left(2\right)$: Whereas an algebraic relationship between $c_s\left(t\right)$ and $m\left(t\right)$ is postulated in the form of a power law,

$\frac{c_s(t)}{c_0}={\left[\frac{m(t)}{m_0}\right]}^{\gamma }$, $\left(2\right)$