## 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.