A) Write the differential equations of motion for vx(t) and vy(t). Solve them.

Here's the solution to :

a) Write the differential equations of motion for $v_x(t)$ and $v_y(t)$. Solve them.

Step 1: Write the differential equations of motion.
From Newton's Second Law, $F=ma$.
For the x-direction, the only force is air resistance:

$$m \frac{dv_x}{dt} = -kmv_x \implies \frac{dv_x}{dt} = -kv_x$$

For the y-direction, forces are gravity and air resistance:

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