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.
Read more about this topic: Liouville's Formula
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