You write $\frac{dv}{dt}$ because it is the form you know best.
The variable in the answer picks the form. Want time, use $\frac{dv}{dt}$; want velocity against displacement, use $v\frac{dv}{dx}$ or $\frac{d}{dx}\!\left(\tfrac12 v^2\right)$.
Next time: Read what is asked before writing the equation of motion.