Let us focus on the velocity only. Discretizing and using the simplest possible approximation for the derivative, we have Euler's forward method for the updated velocity at a time step \( i+1 \) given by
$$ v_{i+1}=v_i+\Delta t \frac{dv}{dt}_{\vert_{v=v_i}}=v_i+\Delta t\left(F_i-\eta v_i-x_i\right). $$Defining a function
$$ h_i(x_i,v_i,F_i)=v_i+\Delta t\left(F_i-\eta v_i-x_i\right), $$we have
$$ v_{i+1}=h_i(x_i,v_i,F_i). $$