Determinant

Wronskian Determinants of Two Functions

We are going to look more into second order linear homogenous differential equations, but before we do, we need to first learn about a type of determinant known as a Wronskian Determinant which we define below.

Definition: Let $f$ and $g$ be two differentiable functions. Then the Wronskian Determinant of $f$ and $g$ is the $2\times 2$ determinant $W(f,g)={\begin{vmatrix}f(x)&g(x)\\f'(x)&g'(x)\end{vmatrix}}=f(x)g'(x)+f'(x)g(x)$.

Sometimes the term "Wronskian" by itself is used to mean the same thing as "Wronskian Determinant". Furthermore, sometimes we can just write "$W$", or "$W(x)$" instead of $W(f,g)$ to represent the Wronskian of $f$ and $g$.

Let's look at some examples of computing the Wronskian determinant of two differentiable functions.

Example 1

Determine the Wronskian of the functions $f(x)=x^{2}$ and $g(x)=3x^{2}$. For what values of $x$ is the Wronskian equal to zero?

We note that $f$ and $g$ are both differentiable functions and that $f'(x)=2x$ and $g'(x)=6x$. Therefore the Wronskian of $f$ and $g$ is:

${\begin{aligned}\quad {\begin{vmatrix}x^{2}&3x^{2}\\2x&6x\end{vmatrix}}=x^{2}\cdot 6x-3x^{2}\cdot 2x=6x^{3}-6x^{3}=0\end{aligned}}$

Therefore the Wronskian of $f$ and $g$ is equal to zero for all $x\in \mathbb {R}$.

Example 2

Determine the Wronskian of the functions $f(x)=e^{x}\sin t$ and $g(x)=e^{x}\cos x$. For what values of $x$ is the Wronskian equal to zero?

We note that $f$ and $g$ are both differentiable functions and that $f'(x)=e^{x}\sin x+e^{x}\cos x$ and $g'(x)=e^{x}\cos x-e^{x}\sin x$. Therefore the Wronskian of $f$ and $g$ is:

${\begin{aligned}\quad {\begin{vmatrix}e^{x}\sin x&e^{x}\cos x\\e^{x}\sin x+e^{x}\cos x&e^{x}\cos x-e^{x}\sin x\end{vmatrix}}=\left(e^{x}\sin x\right)\cdot \left(e^{x}\cos x-e^{x}\sin x\right)-\left(e^{x}\cos x\right)\cdot \left(e^{x}\sin x+e^{x}\cos x\right)\\\quad =e^{2x}\sin x\cos x-e^{2x}\sin ^{2}x-e^{2x}\sin x\cos x-e^{2x}\cos ^{2}x=-e^{2x}(\cos ^{2}x+\sin ^{2}x)=-e^{2x}\end{aligned}}$

Note that $-e^{2x}<0$ for all $x\in \mathbb {R}$ so the Wronskian of $f$ and $g$ is zero nowhere.