Liouville's Formula - Example Application

Example Application

This example illustrates how Liouville's formula can help to find the general solution of a first-order system of homogeneous linear differential equations. Consider

on the open interval I = (0, ∞). Assume that the easy solution

is already found. Let

denote another solution, then

is a square-matrix-valued solution of the above differential equation. Since the trace of A(x) is zero for all x ∈ I, Liouville's formula implies that the determinant

(1)

is actually a constant independent of x. Writing down the first component of the differential equation for y, we obtain using (1) that

Therefore, by integration, we see that

involving the natural logarithm and the constant of integration c₂. Solving equation (1) for y₂(x) and substituting for y₁(x) gives

which is the general solution for y. With the special choice c₁ = 0 and c₂ = 1 we recover the easy solution we started with, the choice c₁ = 1 and c₂ = 0 yields a linearly independent solution. Therefore,

is a so-called fundamental solution of the system.