Computer Algebra Recipes

100 CHAPTER 2. PHASE-PLANE ANALYSIS
PROBLEMS:
Problem 2-27: Harvesting of ¯sh
In suitably normalized units, the e®ect of ¯shing on the normalized population
number x of a single species of ¯sh with a limited food supply can be described
by the following nonlinear ODE,
_x = x (1 ¡ x) ¡ Hx=(a + x);
where H is the harvesting coe±cient and a is a parameter. For H =0,the
remaining ODE is known as the logistic equation. Show that this equation has
an analytic solution and plot the result for a =0:2 andx(0) = 0:1. Discuss the
behavior of the solution.
Then, using the ¯rst-principles RK4 method with step size h =0:1, numerically
investigate this equation as the harvesting coe±cient H is increased from
zero. Plot your results and discuss the change in behavior as H increases.
Problem 2-28: Bucky the beaver
Bucky the beaver attempts to swim across a river by steadily aiming at a target
point directly across the river. The river is 1 km wide and has a speed of 1
km/hr, while Bucky's speed is 2 km/hr. In Cartesian coordinates, Bucky is
initially at (x =1;y = 0), while the target point is at (0; 0). Bucky is initially
swept an in¯nitesimal distance downstream but recovers almost instantly to
continue his swimming motion. His equations of motion are
2 x
2 y
_x = ¡ p ; _y =1¡ p
x2 + y2 x2 + y2 :
(a) Justify the structure of the equations.
(b) Using the ¯rst-principles RK4 method with h =0:01, determine how long
it takes Bucky to reach the target point.
(c) Determine the analytic solution y(x) for Bucky's path across the river.
(d) Plot the analytic and numerical solutions together for Bucky's path.
Problem 2-29: Chemical reaction
Consider the irreversible chemical reaction
2K2Cr2O7 +2H2O+3S! 4KOH + 2Cr2O3 +3SO2, with initially N1 = 2000 molecules of potassium dichromate (K 2Cr 2O 7), N2 =
2000 molecules of water (H 2O), and N3 = 3000 atoms of sulphur (S). The
number X of potassium hydroxide (KOH) molecules at time t sisgivenbythe
rate equation
_X = k (2 N1 ¡ X) 2 (2 N2 ¡ X) 2 (4 N3=3 ¡ X) 3 ;
with k =1:64 £ 10 ¡20 s ¡1 and X(0) = 0. Determine X(t) byusingthe¯rstprinciples
RK4 method with h =0:001. How many KOH molecules are present
at t =0:1 s?att =0:2 s?