Resistive force proportional to velocity 
Find the displacement and velocity of horizontal motion in a medium in which the retarding force is proportional to the velocity.

I kind of understand how to do this problem.
We know that the resistive force $F_r \propto v$. Since $F_r$ is the only force present in the x-direction, Newton's second law gives $$F_r=ma=m\frac{dv}{dt}.$$ My book then says that $F_r=-kmv$. So thus we have $$-kv=\frac{dv}{dt},$$ from which it is trivial to find expressions for $v(t)$ and $x(t)$ by using initial conditions and integration.
The only part about the problem I don't understand is why $F_r=-kmv$. Why does the retarding force depend on the mass $m$? Since $F_r \propto v$, shouldn't we just stick a proportionality constant $k$ in there and have $F_r=-kv$?
 A: Your physical intuition is correct. A resistive force arising from motion in a viscous medium should not depend on the mass of the object. See, for example, Stokes drag for a common model of this kind of resistive force. So it is likely that the force is defined this way to make the equation of motion look nice. If you used a different object with a different mass, $k$ would have to change accordingly. 
A: Force by definition is a change in the momentum of a particle. If the the retarding force is proportional to the velocity, then the correct relationship should be as follows.
, $$\frac{dp}{dt} = \frac{d(mv)}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt} = m\frac{dv}{dt}= -kv$$
Unless you forgot to point out additional information given in the problem
A: In a mass spring damper system it is common to transform the constants $k$ and $d$ using the expressions
 $$ k = m \omega_n^2 \\ d = 2 \zeta m \omega_n $$ where $\omega_n^2 = \frac{k}{m}$. This produces the restorative force as
$$ F_r = -k x - d \dot{x} = -m \omega_n^2 x - 2 \zeta m \omega_n \dot{x} = m \ddot{x} $$
which is simplified by dividing with $m$ to
 $$ \ddot{x} + \omega_n^2 x + 2 \zeta \omega_n \dot{x} = 0 $$
So by scaling the restorative coefficients with mass yields a simplified equation to deal with.
In your case, although not apparent in this section, the author is just using a common convention. The effect of this is that by changing the mass of the problem, the behavior remains the same, or the governing equation does not contain a mass component. 
