Problem: Assume that a Component mass Conservation Equation is given by $\phi \frac{\partial c}{\partial t} + u \frac{\partial c}{\partial x} = -rc$ , where $r=constant$.
Solve the Equation using Method of characteristics.
I know how to solve this problem when it is equal to zero on R.H.S, but struggling to find a solution when $-rc$ is given on R.H.S.
Any ideas?