MA 231: Honors Differential Equations

Do all problems on the blank sheets in order. Please write the problem number in the upper right hand corner of each sheet. Be sure to show all work.

1. Consider the linear system Y' = AY where A is the matrix

(-2 3 0)
(3 -2 0)
(0 0 1 )
a. First compute the eigenvalues for this system.
b. Then sketch the three dimensional phase portrait.

2. In an essay, discuss the behavior of solutions of the forced harmonic oscillator equation

y" + 2 y = cos(w t)

for various values of the parameter w. Be sure to describe all of the different types of cases that occur. Also, provide graphs of ``typical'' solutions in each case as well as a description of what actually happens to the moving mass.

3. Consider the system of differential equations

x' = x(100-x-2y)
y' = y(150-x-6y)
defined only for x,y >= 0.
a. Find all equilibrium points and determine their type using linearization.
b. Sketch the nullclines and indicate the direction of the vector field in between these nullclines.
c. Sketch the phase portrait.

4. Consider the nonlinear system

x' = y
y' = x-x2.
a. Is this a gradient system? If so, find the gradient function.
b. Is this a Hamiltonian system? If so, find the Hamiltonian function.
c. Sketch the phase portrait.