Annuity rate differnetial equation I am trying to find the rate of interest to keep a balance going forever. Initial is 100,000 and withdraw is 8000
So I know I have the general solution
$$p(t) = \frac{8000}{r} + C e^{rt}$$ I know that $p(0) = 100,000$ so I could potentially solve for C but I get lost without an r.
"An initial deposit of 100,000 is placed in an annuity with a bank. What is the minimum interest rate the annuity must earn to allow withdrawals at a rate of $8000/year to continue indefinitely."
 A: It looks as if you are using a continuous model. Let $y(t)$ be the amount in the account at time $t$. Then 
$$\frac{dy}{dt}=-8000+ky$$
for some constant $k$. If we start with $100000$, and want to withdraw money continuously at the rate of $8000$ a year forever, then the smallest $k$ that works is the $k$ that leaves $y$ unchanging at $100000$. 
That gives $-8000+k(100000)=0$, so $k=0.08$.  
That is the continuous compounding rate. The effective annual rate is the $e^{0.08}-1$, about $8.3287\%$.  
I do not know whether you are expected to give the effective annual rate or the the "continuous growth rate." Each is a reasonable answer to the question.
A: This is a very simple way to think about the problem.  Each compounding period, I withdraw $w$ dollars from a balance of $P$ dollars.  Then at the end of a compunding period, the amount I have is
$$(P-w)(1+r) = P$$
if this is to go in perpetuity.  (It could be more, but we are looking for a minimum.)  The interest rate $r$ is then
$$r=\frac{w}{P-w}$$
which in your case is $8/(100-8) = 2/23 \approx 8.7\%$.
