Well looks like the student has abandoned the thread.
This time I can't, not without saying:
Question (b) is more important than the solution to (a).
If the student did question (b), the observations of Ray in post 7 would come to life, rather than being this rather annoying thing that one learns and forgets or it becomes foggy - although one would recall if one had to calculate some numbers from this, and found that you had written the log of a negative number.
This is a (about the simplest) non-linear differential equation. From what the student says she has not done linear differential equations.
It has been said that the whole of Euclid's Elements is meant really to lead up to the Platonic solids. It could be said that the whole of maths teaching for the mass of sciences students is meant to lead the average student up to linear differential equations (with constant coefficients). There, a variety of stuff learned comes together - calculus, complex numbers, linear algebra, trigonometrical and exponential functions, and a lot of physics and engineering etc. For the majority of scientists linear d.e.'s are something indispensable to grasp,
The trouble is the behaviours they predict are mostly quite boring! Certainly with homogeneous d.e.'s There is just one 'stationary' point in the space of the variables, and they just go there and stay there for all time. Maybe they go there extra-boringly, or else they go there by damped oscillations. OK they never quite arrive at the point but after sufficient time they are as close as you please. Or else the stationary point is unstable, and the variable goes boringly off to infinity. In special cases they might oscillate for ever. Okay with several variables one variable can do one thing, another another. In the case of non-homogeneous equations, I think they mostly end up doing what an imposed force is trying to make them do, though in a foot dragging manner. Well I call them boring - they're not that easy and can be a challenge so maybe they are interesting when you really get into them; this I think would mostly be because of the applications.
Whereas nonlinear differential equations have a more varied repertoire. And this is already seen in this simplest of examples. There are two places the system can end up, depending where it starts. It has two stationary states (not quite same thing as previous sentence), one attracting, one repelling. Fairly typical of nonlinear systems the condition of maximum yield it Is also the most unstable condition. If you try to harvest at the theoretical maximum rate, say you had a contract where you had to deliver this same number of fish all the time, you are at risk of losing your whole resource. (I don't know if this typical feature has name, the something principle.)
You do not need to solve the equation to see this. And in most nonlinear differential equations of interest you cannot solve them in terms of the known elementary analytical etc. functions. But you can analyse them qualitatively. In this case a sketch suffices, usually the qualitative will involve some calculation, in which the linear theory is found useful and necessary. Then not so boring after all.
Possibly because of the risk of boring, the teacher in this case has chosen to stimulate with a nonlinear problem.
Maybe the above has become already obvious to the student having done the problem (wasn't sounding so). I will never understand why students don't complete problems (see also my sig.). But I say just getting the answers is inefficient learning. "When you have got the answer the job is not completed." Students can usually get something additional from helpers on this site when they complete, because, underneath, problems are not isolated the way they tend to be treated in the question-and-answer routine, and helpers can point them to building up connected and empowering knowledge rather than just producing answers to questions asked that end there.
