Hill's and Mathieu's equation I am supposed to apply Hill's and Mathieu's equation to parametric pendulum. 


*

*Can you tell me what is the difference between them? 

*Why are they used? 

*What do they describe? 
 A: From wikipedia:
Mathieu equation 
$$\frac{d^2y}{dt^2} + \left[a-2q\cos(2t)\right]y=0$$
is a special form of a Hill equation with only 1 harmonic mode. Meaning the pendulum has only one harmonic or intrinsic frequency of operation (except the basic frequency).
If one replaces the $\left[a-2q\cos(2t)\right]$ term with a more general (periodic) term $f(t)$, one gets the Hill equation. They describe the evolution of a system in terms of its harmonic modes or intrinsic frequencies.
Harmonic modes of a system, are the (intrinsic) frequency modes of a system, when this is analysed in a frequency spectrum analyser. For example when one plays the piano (a physical system as well as musical instrument), when a key is hit, a chord is excited and a sound is produced. The sound has a frequency (lets say for example it is the A note which has a frequency of 440 Hz). 
This is the basic frequency of the (physical) system of the piano chord. But this is not the only one (only for a tuning fork can we say that it exhibits only one frequency). 
Depending on how the chord was hit (for example amount of force applied and direction of force applied), harmonic frequencies (meaning integral multiples of the basic/fundamental frequency) are also present in smaller magnitudes. 
Since the harmonics are integral multiples of the basic system frequency and the basic system frequency depends on the sytem studied, the harmonics also depend on the system studied (in other words they are intrinsic to the system) and the excitation (force) applied. This is in essense what is called Fourier analysis which studies signals and systems in terms of frequencies (meaning periodic functions like sine and cosine).
Etymologicaly, the term "harmonics" relates to "harmony" because the extra frequencies present (the harmonics) usually have a harmonic relation (in music terms) with the basic frequency.
To apply this to a pendulum, you have to have the parameters in place and then solve an ordinary differential equation.
In the equation the $t$ refers to time coordinate, $y$ refers to a coordinate of the pendulum (this can be the height displacement or width displacement or angle, depends on what is modeled). The $a$ and $q$ are extra parameters, $q$ refers to the magnitude of the 1st harmonic (so if $q=0$ there is no 1st harmonic only the basic frequency), while $a$ refers to the basic frequency.
From wikipedia on Hill equation:

Hill's equation is an important example in the understanding of
  periodic differential equations. Depending on the exact shape of f(t),
  solutions may stay bounded for all time, or the amplitude of the
  oscillations in solutions may grow exponentially.[3] The precise form
  of the solutions to Hill's equation is described by Floquet theory.
  Solutions can also be written in terms of Hill determinants.

