Lecture Notes in Differential Equations - Bruce E. Shapiro

243
The Wronskian
In this section we generalize the definition of Wronskian to higher order
equations. If {y 1 , ..., y k } are any set of functions then we can form the
matrix
⎛
⎞
y 1 y 2 · · · y k
y 1 ′ y 2 ′ y ′ k
W [y 1 , ..., y k ](t) = ⎜
⎟ (27.141)
⎝ .
. ⎠
y (k−1)
1 y (k−1)
2 · · · y (k−1)
k
We use the square bracket notation [· · · ] above to indicate that W depends
on a set of functions enclosed by the bracket, and the usual parenthesis
notation (· · · ) to indicate that W depends on a single independent variable
t. When it is clear from the context what we mean we will omit the [· · · ]
and write W(t) where we mean (implicitly) W[· · · ](t).
Denote the general nth order linear differential equation by
L n y = a n (t)y (n) + a n−1 (t)y (n−1) + · · · + a 1 (t)y ′ + a o (t)y = f(t) (27.142)
and let {y 1 , ..., y k } form a fundamental set of solutions to L n y = 0. Recall
that y 1 , . . . , y k form a fundamental set of solutions if they are linearly independent
and every other solution can be written as a linear combination.
If the functions in the W matrix are a fundamental set of solutions to a
differential equation then (27.141) is called the fundamental matrix of
L n y = 0.
The determinant of (27.141), regardless of whether the solutions form a
fundamental set, is called the Wronskian, which we will denote by W (t).
y 1 · · · y k
W [y 1 , ..., y k ](t) =
.
.
= det W (27.143)
∣ y (k−1)
1 · · · y (k−1) ∣
k
Again, we will omit the square brackets [· · · ] and write the Wronskian as
W (t) when the set of functions it depends on is clear as W (t). When the
set of functions {y 1 , ..., y k } form a fundamental set of solutions to L n y = 0
we will call it the Wronskian of the differential equation.
If we calculate the Wronskian of a set of functions that is not linearly independent,
then one of the functions can be expressed as a linear combination
of all other functions, and consequently, one of the columns of the matrix
will be a linear combination of all the other columns. When this happens,