Wronskian Determinants of n Functions

# Wronskian Determinants of n Functions

Recall from the Wronskian Determinants of Two Functions page that if $f$ and $g$ are both differentiable functions, then the Wronskian determinant of $f$ and $g$ is the $2 \times 2$ determinant:

(1)
\begin{align} \quad W(f, g) = \begin{vmatrix} f(x) & g(x)\\ f'(x) & g'(x) \end{vmatrix} = f(x)g'(x) - f'(x)g(x) \end{align}

Furthermore, we can define a Wronskian determinant of $n$ functions $f_1$, $f_2$, …, $f_n$ that are $n - 1$ times differentiable.

 Definition: Let $f_1$, $f_2$, …, $f_n$ all be $n - 1$ times differentiable functions. Then the Wronskian Determinant of $f_1$, $f_2$, …, $f_n$ is the $n \times n$ determinant $W(f_1, f_2, ..., f_n) = \begin{vmatrix} f_1 & f_2 & \cdots & f_{n-1} & f_n\\ f_1^{(1)} & f_2^{(1)} & \cdots & f_{n-1}^{(1)} & f_n^{(1)}\\ \vdots & \vdots & & \vdots & \vdots\\ f_1^{(n-2)} & f_2^{(n-2)} & \cdots & f_{n-1}^{(n-2)} & f_n^{(n-2)}\\ f_1^{(n-1)} & f_2^{(n-1)} & \cdots & f_{n-1}^{(n-1)} & f_n^{(n-1)}\\ \end{vmatrix}$.

When it is clear from the content, we can simply write $W$ instead of $W(f_1, f_2, ..., f_n)$ to denote the Wronskian of $f_1$, $f_2$, …, $f_n$. Furthermore, sometimes we will denote a Wronskian determinant as simply a "Wronskian".

For example, consider the functions $f(x) = x$, $g(x) = x^2$, and $h(x) = x^3$. All three of these functions are twice differentiable. The Wronskian of $f$, $g$, and $h$ is thus:

(2)
\begin{align} \quad W(f, g, h) = \begin{vmatrix} f(x) & g(x) & h(x)\\ f'(x) & g'(x) & h'(x)\\ f''(x) & g''(x) & h''(x) \end{vmatrix} = W(f, g, h) = \begin{vmatrix} x & x^2 & x^3\\ 1 & 2x & 3x^2\\ 0 & 2 & 6x \end{vmatrix} \end{align}

We can then expand this determinant along the first row to get:

(3)
\begin{align} \quad = \begin{vmatrix} 2x & 3x^2\\ 2 & 6x \end{vmatrix} x - \begin{vmatrix} 1 & 3x^2\\ 0 & 6x \end{vmatrix} x^2 + \begin{vmatrix} 1 & 2x\\ 0 & 2 \end{vmatrix} x^3 \\ = (12x^2 - 6x^2)x - (6x)x^2 + (2)x^3 \\ = 6x^3 - 6x^3 + 2x^3 \\ = 2x^3 \end{align}

It's not hard to see that computing Wronskians determinants can become tedious quickly as the number of functions increases. As we'll soon see, Wronskians for $n$ functions $f_1$, $f_2$, …, $f_n$ will play an important role in determining whether the set of linear combinations of $n$ solutions $y = y_1(t)$, $y = y_2(t)$, …, $y = y_n(t)$ to an $n^{\mathrm{th}}$ order linear homogenous differential equation yields all solutions.