Solving a system of Differential Equations: arbitrary constants

Question

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.

Answer 1

To get started, observe that from $$\frac{dp}{dt}=-lp,p(0)=p_0,$$ we can show that $$p(t)=p_0e^{-lt}.$$ (I leave this to you.)

Now, we can rewrite $$\frac{dc}{dt}=lp-kc$$ as $$\frac{dc}{dt}+kc=lp=p_0le^{-lt}.$$ Multiplying this equation by the function $e^{kt},$ which is never zero, we obtain the equivalent $$\frac{dc}{dt}e^{kt}+kce^{kt}=p_0le^{(k-l)t},$$ which we can rewrite as $$\frac{dc}{dt}e^{kt}+c\frac{d\left[e^{kt}\right]}{dt}=p_0le^{(k-l)t}.$$ Applying product rule lets us rewrite this as $$\frac{d\left[ce^{kt}\right]}{dt}= p_0le^{(k-l)t},$$ whence we have for some constant $a$ that $$ce^{kt}=a+p_0l\int e^{(k-l)t}\,dt.\tag{$\star$}$$

At this point, there are two distinct possibilities we must address. First of all, we must address the case that $k=l,$ which lets us show that $c(t)=p_0lte^{-lt}.$ (Again, I leave this to you.)

Now, let's suppose that $k\ne l,$ in which case $(\star)$ becomes $$ce^{kt}=a+\frac{p_0l}{k-l}e^{(k-l)t}.$$ Since $c(0)=0,$ it follows that $a=-\frac{p_0l}{k-l},$ and so we can see that $$c(t)=\frac{p_0l}{k-l}\left(e^{-lt}-e^{-kt}\right).$$

Note: I assumed here that $k,l,$ and $p_0$ were constants. If this is not correct, or if you have trouble demonstrating any of the steps I've shown here, please let me know.