General question about differential equations I have never studied differential equations and have just run across some notes that involve one.  I have a discrete function P that maps points in time (e.g. day1, day2, day3....) onto some real number on the interval [0, 1].
$P(t+1) = (1-3a)P(t) + a(1-P(t))$ 
the notes then define the change in P to be:
$\Delta P(t) = P(t+1) - P(t) = (1-3a)P(t) + a(1-P(t)) - P(t) = a - 4aP(t)$
Ok, fine.  The notes then go on to say let's instead treat time as continuous.  As such the rate of change of this continuous function is:
$ dP(t)/dt =  a - 4aP(t)$  
The notes then state that the solution to this differential equation is:
$P(t) = 1 / 4 + (P(0) - 1 / 4)e^{-4at}$
My question is for what values of $t$ does the above equation hold?  It appears to me from the broader context of this problem that this equation does not in fact hold for any specific $t$ but only for an unfixed $t$ as $t$ increases without bound.  As such, if I wanted to know the value of $P(500)$ in the continuous case I can't actually use this equation.  Is this understanding correct?
 A: Please allow me to use the symbol $n$ for discrete times and the symbol $t$ for continuous times. You consider in fact two different dynamics, one, in discrete time, of a sequence $(P_n)$ such that 
$$
P_{n+1}=a+(1-4a)P_n
$$ 
for every nonnegative integer $n$ and the other, in continuous time, of a function $(Q(t))$ such that 
$$
Q'(t)=a-4aQ(t)
$$ 
for every nonnegative real number $t$. (Yes, I use the letter $Q$ for the solution in continuous time.)
In the case at hand, it happens that both dynamics may be solved explicitly: one has 
$$
P_n=(1-4a)^n(P_0-1/4)+1/4,
$$ 
and 
$$
Q(t)=e^{-4at}(Q(0)-1/4)+1/4.
$$
The similarity is striking (and for good reasons) but one should keep in mind that $(P_n)$ and $(Q(t))$ are different objects and given by different formulas. 
For example it is not true that the sequence $(Q(n))$ of the values of the function $Q$ considered at integer times is $(P_n)$, even if one chooses identical initial conditions $P_0=Q(0)$ (the exception being the common fixed point $P_0=Q(0)=1/4$, yielding $Q(t)=P_n=1/4$ for every nonnegative integer time $n$ and every nonnegative real time $t$).
In the end, if you are interested in $P_{500}$ (the result of $500$ iterations of the dynamics in discrete time), you should use the formula valid in discrete time (the one with a power of $1-4a$) but if you are interested in $Q(500)$ (the result of following the continuous path $t\mapsto Q(t)$ from time $t=0$ up to time $t=500$), you should use the formula valid in continuous time (the one with an exponential).
Hope this answers your question.
Edit 
To solve the discrete time dynamics, a two-steps general technique is first, to look for fixed points, that is, value(s) $b$ such that if $P_n=b$ then $P_{n+1}=b$ (here $b=1/4$), and second, to use a fixed point as the new origin. Here, one considers the dynamics of the sequence $R_n=P_n-1/4$. This reads $R_{n+1}=(1-4a)R_n$ with $R_0=P_0-1/4$. From this formula one should be able to deduce the value of $R_n$ and then of $P_n$.
The same centering-around-a-fixed-point idea works in continuous time: $Q'(t)=0$ means that $Q(t)=1/4$ hence one considers $S(t)=Q(t)-1/4$, which, o miracle, is a solution of the easy-to-solve equation $S'(t)=-4aS(t)$...
