Equilibrium Rates
1.
a) ,
Hence the equilibria (1,1) and (-1,1)
b)
At equilibrium point (1,1),
Since alpha =1 is greater than 0, the solution is unstable , repelling spiral point.
At equilibrium point (-1,1),
Since the eigenvalues are real distinct roots
Trajectories
1.
Since , the general solution is equal to
e)
,
and
f) means trajectories are closed loops. See p. 347 of the text of how to Figure
6.3.3 and Example 4 to draw the phase-plane.
a) (Final Fall 1998 Problem 7) For each of the following answer
AdvancedMathematicsforEngineers(ENG272)
TheCollegeofNewJersey
AnthonyS.Deese,Ph.D.
Quiz#1
Problem#1:Takethefollowingdifferentialequation.
y 6 y + 13 y = 0
Whichofthefollowingsolutionssatisfiesthisdifferential?Showyourwork.
solutionA:y = e3 x sin ( 1 x )
1.
a)
Since alpha = 0 (i.e. the eigenvalue is purely imaginary), the solution is a center and is
stable.
b)
Since the eigenvalues are real distinct roots, with one eigenvalue positive and the other
eigenvalue negative, the solution is unstable.
Eigenvecto
1. A tank contains 10,000 liters of a solution consisting of 100 kg of salt dissolved in water.
Pure rocky mountain spring water is pumped into the tank at the rate of 10 liters/second.
(The mixture is kept uniform by stirring.) The mixture is pumped out
1. (Final Spring 1996 Problem 2) Find the general solution of the following differential
equations.
a)
b)
Solutions
a) Integrating Factor Method:
. Let
b) First find the homogeneous solution and then find the particular solution to find
the general soluti
Consider the Lorenz system below which models "certain modes" of the atmosphere (x, y, and z
are related to pressure, temperature, and humidity)
dx/dt = -10x + 10y,
dy/dt = 25x - y - x z, dz/dt = x y -3 z
a)
Find the equilibria.
b)
Produce a phase-space p