Differential equation involving a time series

[AbusesDimensAnalysis]

Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly.

Any pointers or techniques for solving such things?

 

[fresh_42]

Why is it implicitly? You have $f(n,t)=f(n,0)e^{kt} + e^{kt} -dn^2$ if I saw this correctly (it's a bit tiny for my old eyes). Looks as if you only have to figure out the initial conditions.