The problem is framed as
There are M fish in a pond. The death rate of the fish is constant. The weight of each fish will decrease proportionally to its weight due to its movement and increase proportionally to its weight at the power of $\frac{3}{4}$ due to feed. Build a mathematical model to describe the weight of the fish $w(t)$ and solve it, with the initial condition $w(0) = 1$.
I have formulated the problem as
$\frac{dw}{dt} = g w^{\frac{3}{4}}(t) - kw(t)$
Where $g$ is the constant describing the gain of weight due to feed, and $k$ is the constant describing the loss of weight due to movement.
Assuming that this model is correct (which I appreciate may not be the case), I am not sure how to go about solving it. Writing the RHS as $w(t)(gw^{-\frac{1}{4}}(t)-k)$, I am unable to find a viable partial fraction, since one of the fractions will end up as $\frac{B}{gw^{-\frac{1}{4}}(t)-k}$ where B is some constant.
Any other ideas on how to solve this would be much appreciated.