MA2051  Ordinary Differential Equations
Review For Test II
1.
Work with the following differential equation:

(a)

Find two linearly independent solutions for the
corresponding homogeneous equation.
Prove that your solutions really are linearly independent.

(b)

Find a particular solutionfor the original equation.

(c)

Find the solution for the equation which satisfies
.

(d)

Does the solution have a transient component?
If it does, estimate how long it takes for the
magnitude of this component to become less than .

(e)

Does the equation have a limiting periodic solution? If it does,
identify the amplitude and period of this limiting solution.

(f)

Determine initial values ( and ) so that the
solution to the initial value problem contains no transient
component.
2. Consider the two functions

(a)

Find a linear, constantcoefficient, homogeneous
differential equation that has and as solutions.

(b)

Determine initial conditions and
so that is the solution for the differential
equation that you found in (a).
3.
A mass of 6 kg is suspended from a spring that extended m
when a 1 kg mass was attached. Assume that friction damping
can be ignored (until part (d)). You pull the mass
down5 cm and release it from rest.

(a)

Write down the mathematical model for this
system. Use the given data to
specify the numerical values for the parameters in the model.

(b)

Determine the natural frequency,the
period,and the amplitude
of the motion of the mass.

(c)

How much would you have to increase the mass in order to
double the period? How much would you have to
increase the mass in order to double the the amplitude?

(d)

Take the same springmass system and immerse it
in a vat of molasses. It is no longer reasonable to
ignore the force of friction damping.
For what values of the
damping coefficient p will the spring still exhibit
oscillations about the steady state? Explain your answer.
4. Consider the following system of firstorder
differential equations:

(a)

Find two solution pairs and verify that they are linearly independent.
Give the general solution for the system.

(b)

Find the solution satisfying initial conditions
and .

(c)

Graph the solution obtained in part (b) in the
plane, you would find that it is a spiral going
out away from the origin.

(d)

Using only the differential equation (and not the actual solution),
show that the quantity is actually increasing
along any solution curve and so the solution must be heading
away from the origin.
5.
Use two steps of Euler's method to approximate the solution to
at t = 1. Compare your numerical result with the exact solution
obtained in problem #1 above. What is the error?