Differential Equations: Math 3226
Problem 1. Solve
Problem 2. Consider the initial-value
Using Euler's method,compute three different approximate solutions
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
Problem 4. Consider the system
- Sketch the nullclines and find the direction of the vector
field along the nullclines.
- 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
- Describe the behavior of solutions near the equilibrium points.
Problem 5. Consider the system
- Find the value of a which gives a system with repeated or
- Consider the system with the value of a found in 1. Find
the general solution.
- 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.
- Write the corresponding second order equation and first
- Determine all values of kd at which a bifurcation occurs; and
- Give the general solution of the system when kd = 1,and kd=2.
Problem 7. Consider the system
- Find the equilibrium points.
- Classify the equilibrium points as: source, sink, center,
and so on...
- Sketch the phase-plane near the point (0,0).
Problem 8. Find the solution to the initial value problem
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