Following is an example of a math mini-project. Students were required to use Scientific Notbebook (or other computer algebra system) to explain an application of differential equations.

Once
I saw a girl in a garden playing with the swing . Before she start to play ,
her father gave a certain velocity to the swing.And the swing started moving
on and on and finally it stopped . If we know the numerical values of initial
velocity and mass of the the girl and the swing, we can find out the movement
of the swing with respect to time. The movement of the pendulum came into my
mind.But the pendulum of a clock seems never stop.

The
movement of the pendulum and that of the swing are pretty similar and we can
find out their displacemets with respect to time by using differential
equation.

To find the movement of the pendulum

Problem description:

We find out that when the pendulum is in the vertical position , the maximum
point where the

pendulum can reach is
metre and the constant
is found out to be

From there we can get the necessary information to solve the problem.

A simple harmonic equation can be described as

,
Exact solution is:

The movement of the
pendulum

To
find the movement of thd swing

Problem
description:

The weight of the girl and the swing is
.
And the initial velocity is
And
the air resistance is
times of the velocity and frictional loss constant
is
The
movement started from
of the original position .(All the values are assumed approximately)

Solution:

we can set the equation by using Newton's Law of motion which states that the
external force applied is equal to the product of the mass and the
acceleration.

where
is

,
Exact solution is:

The movemet of the
swing

Conclusion:

For the
pendulum case, it is moving continuously and it seems never stop.After solving
the equation
we find out that its movement is like a cosine waveform and it is oscillating.

In the
case of the swing , the swing start moving and it speed become slower and
slower and finally it stops.And the waveform is

It
means that the sum of sine wave and cosine wave decrease exponentially.

We find
this kind of problems in our daily life . If we have a better understanding in
how to solve differential equations using the steps used above,and it is a
marvel of life.

Bibliography

1) Basic Technical mathematics with calculus by Allan J Washinton