Math 221  Project with Maple  Due on April 6, 2001

Name: Score:

Suggestions: To get started, go to my web site http://www.math.unl.edu/∼bdeng to get the ode sample file in Maple.

1 (a) Use Maple to find the exact solution to the logistic equation

\frac{dP (t)}{dt} = rP (t)(1 − \frac{P (t)}{K})

with r > 0 the intrinsic growth rate and K > 0 the carrying capacity.

(b) Plot the slope field for the equation with r = 0.5 and K = 5 together with solutions over the time interval [0, 20] with distinct initial conditions P (0) = 1, 2, 3, 4, 5, 6, 7, 8.

2 (a) Use Maple to find the exact solution to the system of equations

x′(t) = y(t), y′(t) = −2x(t) − y(t).

(b) Use Maple to find the solution that satisfies the initial conditions x(0) = 1, y(0) = −2. Plot the both x(t), y(t) together over the time interval 0 ≤ t ≤ 20.

(c) Use Maple to find the solution that satisfies another initial conditions x(0) = −2, y(0) = −1. Plot the x component of this solution and that of the solution of (b) together over the time interval 0 ≤ t ≤ 20.

3 (a) Consider the equations of two competing species

\begin{aligned}\frac{dx}{dt} & = x(1 − x − 0.5y) \\ \frac{dy}{dt} & = y(2 − 1.5y − x). \end{aligned}

(i) Plot the direction field similar to Figure 9.4.1. of the supplement. (ii) Plot the direction field together with several solutions in one plot. Such a plot

is referred to as a phase portrait of the equations. With the exception of the direction field, this plot should be similar to Figure 9.4.2 of the supplement. (You need to include a similar number of trajectories as in Figure 9.4.2 in your plot.)

(iii) Plot the solutions curves x(t), y(t) verses the time t over the interval [0, 20] with initial conditions x(0) = 0.3, y(0) = 0.08. You need to put these two curves in one plot similar to Figure 9.5.3.

THE END