Suppose that when a certain lake is stocked with fish, the birth and death rates $$ and $$ are both inversely proportional to $\sqrt[\phantom{2}]{P}.$

$($a$)$ Show that $P(t)=(\frac{1}{2}kt+\sqrt[\phantom{2}]{P_0}{)}^2,$ where $k$ is a constant. $.$

$($b$)$ If $P_0=121$ and after $8$ months there are $225$ fish in the lake, how many will there be after $16$ months?

$($a$)$ Find the differential equation $\frac{dP}{dt}$ where $=\frac{a}{\sqrt[\phantom{2}]{P}}$ and $=\frac{b}{\sqrt[\phantom{2}]{P}}.$

$\frac{dP}{dt}=$