I don't have much time today, but I can whip out a solution for a).Originally Posted bykingkaisai2

Since thus where x is a constant.

So where k and x are constant.

The general solution of such a linear differential equation is of the form

So

Plugging these into your differential equation gives:

Comparison shows that .

So far we have:

Now, we know that the maximum charge on the battery is Q. Thus the limit of q(t) as t goes to infinity is Q. Note that b is a negative number (given the range of acceptable x values.) Thus the limit of the exponential function as t gets very large is 1. ie.

Thus and we know according to the problem statement. Thus A = Q and your final solution is:

-Dan