For a research project I am carrying out I am required to solve the system:
$\frac{dp}{dt} = -lp $, $ \frac{dc}{dt} = lp - kc $ with initial conditions $p(0) = p_0 $ and $c(0) = 0 $. Here, $p,c$ denote concentrations and $l,k$ are reaction rates.
I believe that I have solved the system to reach a solution $$ c(t) = (\alpha + \frac{\beta}{l - k})e^{-kt} - \frac{lp_0}{l-k}e^{-lt} $$ where $\alpha, \beta$ are arbitrary constants.
I am quite sure (barring any typos!) that this solves the system however I can't seem to get rid of the two arbitrary constants. I don't know if I need another initial condition or am missing some relation between the two constants?
Since this is a Pharmacokinetics problem, I was able to find a relation between $ \alpha$ and $\beta$ by considering the half life. However, this resulted in something like:
$$ \alpha = \frac{((lp_0)^p - \beta^k)^{1/k}}{l-k} $$
which didn't seem to help.
Any help or feedback would be appreciated.