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**mr fantastic** I get $\displaystyle v = \frac{mg - (mg - k v_0) e^{-kt/m}}{k} = \frac{mg}{k} - \left(\frac{mg}{k} - v_0 \right) e^{-kt/m}$.

The DE is $\displaystyle \frac{dv}{dt} = \frac{mg - kv}{m} \Rightarrow \frac{dt}{dv} = \frac{m}{mg - kv}$.

The DE clearly suggests a terminal velocity $\displaystyle v_t = \frac{mg}{k}$, which is consistent with my solution but obviously not yours (your solution has v --> +oo as t --> +oo).

At t = 0 my solution gives $\displaystyle v = v_0$ whereas your solution gives $\displaystyle v = -v_0$ ....