Drag

A ball of mass $m$ is thrown and experiences a frictional drag force as it moves through a fluid. The forces acting on this ball are gravity $-mg\hat{z}$ and the drag force. The drag force has the form,

$\large \vec{F}_{drag} = -a\vec{v} - b\vec{v}|\vec{v}|,$

where $a$ and $b$ are constants. For a low Reynolds number, the linear term $-a\vec{v}$ usually dominates whereas for a high Reynolds number, the quadratic term $-b\vec{v}|\vec{v}|$ dominates.

If the ball falls for a long time, the velocities in the $x-$ and $y-$directions will approach zero while the velocity in the $z-$direction will approach a constant called the terminal velocity when the drag force balances the gravitational force, $-mg-av_z - bv_z|v_z|=0$. Solving this equation for $v_z$ yields the terminal velocity.

For a certain case: $m=$  kg $a=$  N s/m $b=$  N s²/m²
At $t=0$:  $x=0$  m, $y=0$ m, $z=0$ m,  $v_x=$  m/s,  $v_y=$  m/s,  $v_z=$  m/s.

What equations would have to be put into the differential equation solver to determine the trajectory of the ball?