1. Hunter Rose, R10572598
10/20/2014
Differential Equations
Harmonic Oscillator
A harmonic oscillator is a special case where, when a systemis displaced from its
equilibrium conditions, it will experience a resultant force that is proportional to its
displacement; x. Harmonic oscillators can be used to find displacement as a function of time as
well. This will relate to us in Differential Equations in multiple ways, two of which are in relation
to Hooke’s law and Newton’s second law, as well as almost anytime we encounter a spring or
something similar to one.
Harmonic oscillators appear in numerous places in the world including pendulums,
masses connected to springs, and acoustical systems. One of the simplest harmonic equations
we will see is y’’+ky=0, which is a simple harmonic function, otherwise known as free un-
damped motion. An un-damped harmonic function is one in which your force or you (ky) is the
only thing acting on your system. If this is not the case then you encounter what is called a
damped harmonic function in which there are more forces acting on your system, such as
friction. Dampening can be broken up into different sub categories such as over-damped in
which the function will decay to the equilibrium positon without oscillating, underdamped in
which the frequency and size of the oscillations decreases with time, or possibly critically
damped.
We are looking at the case of a simple harmonic oscillator in which there is no
dampening occurring. In this scenario a Laplace transformation could be used to solve an initial
value problem involving a harmonic oscillator equation. For example, in our book on page 233
there is an example of solving an initial value problem that describes the forced, un-damped
and resonant motion of a mass on a spring, which is a simple harmonic oscillating equation
x’’+16x=cos(4t). After using a Laplace transformation on this problem you will find that
x(t)=(1/4)sin(4t)+(1/8)t*sin(4t). Another example of using a Laplace Transformation on an Initial
value harmonic oscillator problem is question number 14 in section 4.6 of our book. This
question asks you to derive a systemof differential equations describing the motion of some
springs then use a Laplace transformation to solve them, which will be a pretty hefty amount of
work but can be done.

2. References
 Advanced Engineering Mathematics, Fifth Edition, by Dennis G. Sill and Warren S.
Wright.
 NCSU department of mathematics slides on harmonic oscillator equations
http://www.ncsu.edu/crsc/events/ugw05/slides/root_harmonic.pdf
 http://scipp.ucsc.edu/~haber/ph5B/sho09.pdf
 http://en.wikipedia.org/wiki/Harmonic_oscillator#Simple_pendulum