Obtaining the Mathieu equation

We want to rewrite

as

and determine how a, q, and x depend on the original m, k, U, V, w, and t.

To do that we need to

1. modify the cosine argument from wt to 2x and work out the effect on the other occurrences of t in the original expression
2. collect the constants in front of the second part of the equation

To modify the argument of the cosine function from wt to 2x we need to equate the two. Rearrangemenet gives that one can use the substitution t = 2x/w, or

Since the differential equation has the second derivative to t in it, we now need to find how this second derivative to t is related to the second derivative to x.

First we find how the first derivates relating by using the chain rule. We can write, in operator notation,

This result we can use to obtain the expression of the second derivative to t as that of a second derivative to x, as follows:

Substituting this operator and using t = 2x/w, we now can write our original expression as

By dividing by 4/mw² we can collect the constants in front of the second part of the equation

and comparison with the target expression we see that we need, in addition to

the following substitutions are required

to rewrite our initial equation as

Note
because in the derivation we divided by w² we are not allowed to remove the cosine term by setting w to zero, but set q to zero in stead.