Differential Equations: Math 3226

Problem 1. Solve

Problem 2. Consider the initial-value problem

Using Euler's method,compute three different approximate solutions corresponding to , , and over the interval . Graph all three solutions. What predictions do you make about the actual solution to the initial-value problem? How do the graphs of these approximate solutions relate to the graph of the actual solution? Why?

Problem 3. The graph of f(y) is given.

Sketch the phase line and some solutions for the differential equation dy/dt=f(y).

Problem 4. Consider the system

1.
Sketch the nullclines and find the direction of the vector field along the nullclines.
2.
Show that there is at least one solution in each of the second and fourth quadrants that tends to the origin as . Similarly, show that there is at least one solution in each of the first and third quadrants that tends to the origin as .
3.
Describe the behavior of solutions near the equilibrium points.

Problem 5. Consider the system

1.
Find the value of a which gives a system with repeated or double eigenvalues.
2.
Consider the system with the value of a found in 1. Find the general solution.
3.
Find the particular solution which satisfies

Y0 = (1,1)

Problem 6. Consider the harmonic oscillator with Mass m=1, spring constant ks = 1, and coefficient of damping kd.

1.
Write the corresponding second order equation and first order system,
2.
Determine all values of kd at which a bifurcation occurs; and
3.
Give the general solution of the system when kd = 1,and kd=2.

Problem 7. Consider the system

1.
Find the equilibrium points.
2.
Classify the equilibrium points as: source, sink, center, and so on...
3.
Sketch the phase-plane near the point (0,0).

Problem 8. Find the solution to the initial value problem

where