Differential Equation - Word Problem Need help with this problem. I want to write it in a differential equation of the form:
$$p(t)' + f(t) p(t) = g(t)$$
Because of restoration of an island habitat, the maximum population of birds it can support at a time t is given by $1100*\exp(t/80)$. The growth rate of the population $p(t)$ is equal to 1/20 of the difference between the maximum population and the current population. Initially the island has a population of 200 birds.
 A: Since the rate of growth $p(t)'$ is equal to $1/20$ of the difference of max and current population
$$p(t)'=\frac1{20}\bigg(1100\exp(t/80)-p(t)\bigg)$$
and by simple manipulation
$$p(t)'+\frac1{20}p(t)=55\exp(t/80)$$
with the initial condition
$$p(0)=200$$
A: When you get a differential equations word problem, key phrases to look for include "growth rate" and "current ."  "Growth rate" corresponds to the first derivative, and the "current __" corresponds to the actual function.
So, as we read this problem, we can note "The growth rate... is equal to," so we write down:
$$p'(t) = \text{(to be filled in later)}$$
Continuing, we see "1/20 of."  "Of" in math word problems often refers to multiplication.  So, we write:
$$p'(t) = \frac{1}{20}\cdot\left(\text{to be filled in}\right)$$
Continuing again: "difference of maximum population and the current population."  Well, we're given the maximum population: $1100 \exp(t/80)$.  What is the current population?  That's $p(t)$.  So, filling in our equation again:
$$p'(t) = \frac{1}{20}\cdot\left(1100 \exp\left(\frac t {80}\right) - p(t)\right)$$
Finishing the problem statement, we find that the original population is $200$ birds.  As you wrote in the comments, yes, this is the initial condition: $p(0) = 200$.  So, we're done writing the differential equation.  To make the equation fit the form you want, we can do some algebra:
$$\begin{align}p'(t) = \left(\frac{1}{20}\right)1100 \exp\left(\frac t {80}\right) - \frac{1}{20}p(t)&,\qquad p(0) = 200\\
p'(t) + \frac{1}{20}p(t) = 55 \exp\left(\frac t {80}\right)&,\qquad p(0) = 200\end{align}$$
