# 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

*Y*_{0} = (1,1)

**Problem 6.** Consider the harmonic oscillator
with Mass *m*=1, spring constant *k*_{s} = 1, and coefficient
of damping *k*_{d}.

- 1.
- Write the corresponding second order equation and first
order system,
- 2.
- Determine all values of
*k*_{d} at which a bifurcation occurs; and
- 3.
- Give the general solution of the system when
*k*_{d} = 1,and *k*_{d}=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

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