38. Wronskian determinant. Liouville formula.
Wronskian determinant
Given
functions
,
then the Wronskian determinant
is the determinant of the square matix
where f(k) indicates the k th derivative of f (not exponentiation).
The
Wronskian of a set of functions
is another function, which is zero over any interval where
is linearly dependent. Just as a set of vectors is said to be
linearly dependent when there exists a non-trivial linear relation
between them, a set of functions
is also said to be dependent over an interval
when there exists a non-trivial linear relation between them, i.e.,
for
some
,
not all zero, at any
Therefore the Wronskian can be used to determine if functions are independent. This is useful in many situations. For example, if we wish to determine if two solution of a second- order differential equation are independent, we may use the Wronskian.
Consider the functions x2, x , and 1. Take the Wronskian:
Note that W is always non-zero, so these functions are independent everywhere. Consider, however, x2 and x:
Note that W is always non-zero, so these functions are independent everywhere. Consider, however, x2 and x:
Here W is always zero, so these functions are always dependent. This is intuitively obvious, of course, since
2x2+3=2(x2)+3(1)
Given n linearly
independent functions 
,
we can use the Wronskian to construct a linear differential equation
whose solution space is exactly the span of these functions. Namely,
if g satisfies
the equation;
then
for some choice of
As
a simple illustration of this, let us consider polynomials of at most
second order. Such a polynomial is a linear combination of
and
.
We have
Hence,
the equation is
which indeed has exactly polynomials of degree at most two as
solutions.
Liouville formula.
Definition.
Let
are
real-valued functions on
,
which are
times differentiable on
.
Then their Wronskian is defined by
Theorem
(The
Liouville
formula) Let
be a sequence of
solutions of
is continuous. Then the Wronskian
of this sequence satisfies the identity
for
all
.
Recall
that the trace
trace A of the matrix A is the sum of all the diagonal entries of the
matrix.
Proof.
Let the entries of the matrix (
then
We use the following formula for differentiation of the determinant, which follows from the full expansion of the determinant and the product rule:
Indeed,
if
are real-valued differentiable functions then the product rule
implies by induction
Hence, when differentiating the full expansion of the determinant, each term of the determinant gives rise to n terms where one of the multiples is replaced by its derivative. Combining properly all such terms, we obtain the derivative of the determinant is the sum of n determinants where one of the rows is replaced by its derivative, that is, (1).
The
fact that each vector
satisfies the equation
can be written in the coordinate form as follows
