# An application with partial fractions and separable equations

## Homework Statement

Suppose that a town has a population of 100,000 people. One day it is discovered that 1200 people have a highly contagious disease. At that time the disease is spreading at a rate of 472 new infections per day. Let N(t) be the number of people (in thousands) infected on day t.

1. Explain why the model dN/dt = kN(100-N) might be a good model for the spread of the disease
2. Use partial fractions to solve the differential equation and find the function N(t).
3. According to this model if the disease continues unchecked when will the number of people infected reach 30,000.
[suggestion: solve for the constant k before you do any integration]

## The Attempt at a Solution

From the equation given in part 1, I plugged in the values give:

472 = k(1.2)(100-1.2)

and found k = 3.98. From there I separated the problem and integrated:

∫1/(N(100-N))dN = ∫3.98dt

using to partial fractions to solve, I found this to reduce to:

(1/100)ln|N| + (1/100)ln|100-N| = 3.98t + C
ln|N| + ln|100-N| = 398t + C
N + 100 - N = e^(398t + C)

But the N's cancel out and I cant find the mistake i have made. Any help would be great.

## Answers and Replies

I didn't follow all of your working step by step, I just noticed that the very last line of your calculation isn't right. ln(a) + ln(b-a) != ln(b). Rather, ln(a) + ln(b-a) = ln(a(b-a)). In other words, when you exponentiate everything on the left hand side, the N term shouldn't cancel.