1. Clicker Question 1
What is the general solution to the DE dy / dx = x(1+ y2)?
A. y = arctan(x2 / 2) + C
B. y = arctan(x2 / 2 + C)
C. y = tan(x2 / 2) + C
D. y = tan(x2 / 2 + C)
E. y = sec(x2 / 2) + C

2. Clicker Question 2
What is the specific solution of the DE dy / dt = 4y2 for which y = 1 when t = 1?
A. y = e4t + 1 – e4
B. y = 1 / (5 – 4t)
C. y = 1 / (4t – 3)
D. y = -1 / 4t + 5/4
E. y = 1 / 4t + 3/4

3. Application of DE’s: Population Growth (10/21/13)
Let P be the size of a population and let t be time. For example, if the population grows at a rate proportional to its size, this say that it satisfies the DE: dP / dt = k P , k being the relative growth rate.
This is separable, and we know the general solution is P = A e kt where A is the starting population.
This is, naturally, called exponential growth.

4. The Logistic Model of Growth
Many populations may grow exponentially at first, but eventually that growth rate slows as capacity (space, food, etc.) is reached.
That is, as time passes, k will approach 0.
If the maximum capacity of the population is denoted M, a simple expression which approaches 0 as P approaches M is – P / M .

5. The Logistic DE
Thus a DE which would model this “exponential growth at first but slowing of the growth rate as P approaches its maximum capacity” would be

6. Example
Suppose a population growing by the logistic model has a maximum capacity of 1000 and displays an initial growth rate of 8%.
Look at an Euler’s Method approximate solution assuming an initial population of 2.
Can we explicitly solve this DE? Is it separable?

7. Assignment for Wednesday
Do the problems handed out in class. These are not to hand in.
Read Section 9.4 through page 610.