## The Wronskian

The Wronskian of functions
$f,g$
at a point
$x_0$
is the determinant of the matrix
$W(f,g,x_0)= \left| \begin{array}{cc} f(x_0) & g(x_0) \\ f'(x_0) & g'(x_0) \end{array} \right| =f(x_0)g'(x_0)-g(x_0)f'(x_0)$

Typically we consider the Wronskian when analysing the solutions of differential equations. Suppose we have the solutions
$f,g$
to a differential equation. If the Wronskian is zero at some point then the solutions are dependent.
Example:
$f(x)=sin x , \: g(x)=cos x$
are solutions to the equation
$y''+y=0$

$W(f,g)= \left| \begin{array}{cc} sin x & cos x \\ cos x & -sin x \end{array} \right| =-cos^2 x - sin^2 x=-1$

Hence
$sin x, \: cos x$
are independent or fundamental solutions.