I have a differential equation inspired by a population dynamics problem and I'm don't know how to solve it. I wonder if anyone here can tell me if a closed form solution exists and how to find it.
$$ \frac{\partial u}{\partial t} = p(t)u(t) - k $$
Where $p$ is the concentration of (non-replenishing) food, $u$ is a population, and $k$ is the rate at which members of the population die. Since $p$ is non-replenishing,
$$ p(t) = p_0 - \int^t_0 u(\tau) d\tau $$
Combining these equations gives the differential equation
$$ \left( p_0 - \int^t_0 u(\tau) d\tau \right)u(t) - \frac{\partial u}{\partial t} - k = 0 $$
How do I solve this?
Assuming you make the death rate into a real rate, you can write your equations as $$ \dot u=pu-ku\\ \dot p=-u $$ and by differentiating the second the first equation becomes $$ -\ddot p = -p\dot p+k\dot p $$ which can be integrated to $$ \dot p =\frac12 p^2 - kp + C $$ which can be solved as the $\arctan$ or $\text{Artanh}$ depending on $C$.