Floquet's Theorem

Floquet's theorem states that a solution to a linear differential equation with periodic coefficents, with a periodicity of t, can be expressed as

where f(x) is a periodic function with period t.

The following proof has been adapted from "The Mathematics of Physics and Chemistry," by H. Margenau and G.M. Murphy, van Nostrand and Company, Inc., New York, Second Edition, 1956, pg 80-81.

A general linear differential equation can be written as

If u1 and u2 are two linearly independent solutions of this equation, a particular solution may be written as the linear combination of the two:

u = A1 u1 + A2 u2

If the coefficients fn(x) are periodic functions, with periodicity t, substitution of x by x+t will not affect the differential equation. This does not mean that u1(x+t) = u1(x) and u2(x+t) = u2(x) , but that the values at x+t can be written as linear combinations of the u1 and u2 values at x, i.e,

u1 (x+t) = a11 u1(x) + a 12 u2(x)
u2 (x+t) = a 21 u1(x) + a 22 u2(x)

Using these expressions we can write for the particular solution, u:

since the A1 and A2 can be choosen, while the as are fixed by the choice of u1 and u2, we can introduce a constant k, such that it obeys the following two equations:

so that we can substitute this our previous equation to give

which means that a particular solution to the differential equation exists, such that if x is increased by t, the value is multiplied by a constant k. If k were 1, the solution would be periodic.

The value of k is restricted by the equations that allowed us to introduce it in the first place, but we can choose it to be

where now m is a constant that depends on the choice of u1 and u2.

At the same time we choose the solution u(x) to be the following function of f(x)

From this choice of u(x) we can write

while at the same from the previous derivation,we can write

we find that f(x) must a periodic function with periodicity t, since the two equations can only be equal if

f(x+t) = f(x)

Thus we found a particular solution to a linear differential equation with periodic coefficients with a periodicity of t can be written as

where f(x) is a periodic function with the same periodicity t.

© 1998, University of Richmond