NOTE: revised equations in problem 5 on th 4/21
demonstration (follow along)
- (a) plot phase portrait and a few
trajectories for system x' = 3x, y' = y
phaseport = StreamPlot[{3x, y}, {x, -5, 5}, {y, -5, 5}]
traj1 = ParametricPlot[{Exp[3 t], Exp[t]}, {t, -10, 5}, PlotRange -> 5]
traj2 = ParametricPlot[{5 Exp[3 t], - 2 Exp[t]}, {t, -10, 5}, PlotRange -> 5]
traj3 = ParametricPlot[{- Exp[3 t], - 4 Exp[t]}, {t, -10, 5}, PlotRange -> 5]
Show[phaseport, traj1, traj2, traj3]
(b) describe the eventual behavior of the system
(c) sketch-guess by hand possible shapes of solutions for x, y using only phase
portrait for the following sets of initial conditions: x(0)=1 and y(0) = 0;
x(0) = 0 and y(0) = 1; x(0) = 1 and y(0) = 4.
(d) compare guesses with actual solutions.
- (a) plot phase portrait for pendulum y'' = -sin y
given as system: x' = -sin y, y' = x
(b) describe the eventual behavior of the system
for homework
these problems are a mix of mathematica exercises and problems to do by
hand. for the mathematica part, include printouts of your mathematica
worksheets, with code and graphs clearly labeled with the associated
problem number.
the due date will be listed on the homework page.
though you may get help
from me or your classmates, all of your code should be typed in (not copied) by
yourself.
- (example from lecture). consider the system x'=2x+y, y' = x+2y.
(a) plot the phase portrait.
(b) describe the eventual behavior of the system.
(c) use the phase portrait to sketch-guess by hand shapes of the solutions
x, y for the following sets of initial conditions: x(0) = 1 and y(0) = 0;
x(0) = -1 and y(0) = 0; x(0) = y(0) = 1.
(d) compare your sketch-guesses in (c) with the actual graphs
using the solutions we found in lecture:
x = c_1 e^(3t) + c_2 e^t, y = c_1 e^(3t) - c_2 e^t
- consider the system x' = x+y, y' = 4x+y.
(a) plot the phase portrait.
(b) describe the eventual behavior of the system.
(c) find the general solution to the system using the method from lecture.
(d) graph the solutions x(t) and y(t) for the following sets of initial
conditions: x(0) = 1 and y(0) = 0; x(0) = 1 and y(0) = 2; x(0) = 0 and
y(0) = -1.
- consider the system x' = x-2y, y' = y.
(a) plot the phase portrait.
(b) describe the eventual behavior of the system.
(c) find the general solution to the system by reducing to a first-order
single ODE for x.
(d) graph the solutions x(t) and y(t) for the following sets of initial
conditions: x(0) = 1 and y(0) = 0; x(0) = 2 and y(0) = 1; x(0) = 0 and
y(0) = 1.
- (mixing with two tanks example from lecture)
consider the system x' = -0.4x + 0.1y + 3, y' = 0.4x - 0.4y.
(a) plot the phase portrait.
(b) describe the eventual behaviour of the system.
(you may assume x, y >=0, and may need to vary the range of your portrait.)
are there equilibrium solutions? if so, find them.
- (predator-prey example from lecture, with modified coefficients)
consider the system
x' = 2x - 0.1xy, y' = 0.005xy - 0.3y.
here x(t) represents
the rabbit population and y(t) represents the fox population.
(a) plot the phase portrait.
(b) describe the eventual behaviour of the system.
(you may assume x, y >=0, and may need to vary the range of your portrait.)
are there equilibrium solutions? if so, find them.
(c) interpret your answer for (b) to determine for what initial populations
(at least roughly) the rabbit or fox populations will all die off.
help
see the lab 1 and
lab 2 pages for reminders on how to do some things
in mathematica and pointers to documentation.
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