Example Application
This example illustrates how Liouville's formula can help to find the general solution of a firstorder 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 squarematrixvalued 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 c_{2}. Solving equation (1) for y_{2}(x) and substituting for y_{1}(x) gives
which is the general solution for y. With the special choice c_{1} = 0 and c_{2} = 1 we recover the easy solution we started with, the choice c_{1} = 1 and c_{2} = 0 yields a linearly independent solution. Therefore,
is a socalled fundamental solution of the system.
Read more about this topic: Liouville's Formula
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