1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 137

Numerically compute and graph the solution to this initial value prob-

lem on the interval [O, 25] for Po = 75, a = .15, and (i) b = .03 and

(ii) b = - .03. What is limt_, 00 P(t) in each case?

7. The following two initial value problems are variations of the logistic
   law model. Compute and graph numerical solutions to these problems
   on the interval [O, 25]. However, before computing a solution see if you
   can guess whether the population will increase or decrease in each case.

dP _ 2 _

a. dt - .2PlnP - .02P , P(O) = 1

dP _ .2P 2 _

b. dt - ln p - .02P , P(O) = 2

8. In 1963, F. E. Smith proposed the following model to explain the pop-
   ulation growth of a species of water fleas under laboratory conditions.

dP

dt

. 2P-.02P^2

1 + .OlP '

P(O) = 1.

Use SOLVEIVP or your computer software to compute and graph the

solution of this initial value problem on the interval [O, 25]. How does

the solution behave as time increases? Does limt_, 00 P(t) exist? If so ,

what is it?

9. Another proposed modification of the logistic law model is

dP c

dt = P(a - bP)(l - p); P(O) =Po

where a, b, c > 0. Compute and graph a numerical solution to this initial

value problem on the interval [O, 25] for a = .3, b = .005, c = .02, and

Po= 100.

10. Compute and graph numerical solutions of the following two initial value
    problems on t he interval [O, 25]. Before computing the solutions see if
    you can guess the behavior of the solution curves. Do they increase?
    Do they decrease? Do they increase and then decrease? Or, do they
    decrease and then increase? Does limt_, 00 P(t) exist? If so , what is it?

dP 100

a. dt = .15Pln( p ); P(O) = 25

dP 100

b. dt = -.15P(l - p ); P(O) = 25