Simple Modelling Question (ODE, 1st-order, population growth)

I'm looking at MIT's open course-ware class on differential equations and reading through the first problem set, found here: http://ocw.mit.edu/courses/mathemati..._03S10_ps1.pdf. First, I've only done math. I've never had *any* science classes (except in high school, and at that the classes basically just told you that an atom has a positron or whatever).

Question 1: So while I got 0.a right, the answer sheet says that the units of k(t) are time^-1. WTF does that mean? Just that k(t) is a function of time? I would have thought that the units of k(t) would be, say, (kilo-oryx)/time (rate of change of thousands of oryx over whatever unit of time).

Question 2: k(t) is the growth rate of the population as time varies, right? So I would have thought that the answer to 0.b would be just x' = (k_0)/(a + t)^2, since x'(t) is also the rate of change of the population as time. But the answer sheet says it's x' = (x*k_0)/(a+t)^2. Another WTF?

[Edit: Note the question I'm referring to starts at the top of the second page of the (.pdf) .]