Wronskian

The Wronskian is a convenient method to determine linear independence or -dependence across a set of functions such as

$$f\_1(x),...,f\_n(x)$$

It is used in the process of solving nonhomogenous differential equations with the universal Method of Variation of Parameters.

Workflow

The Wronskian of a set of functions is denoted as $W(f\_1(x),...,f(\_n(x))$ and is the determinant of the $n\text{*}n$ matrix. $n$ is given by the amount of functions.

$$W(f\_1,f\_2,\dots ,f\_n)(x)=\left|\begin{array}{cccc}f\_1(x)& f\_2(x)& \cdots & f\_n(x)\\ f\_1^{\prime }(x)& f\_2^{\prime }(x)& \cdots & f\_n^{\prime }(x)\\ ⋮& ⋮& \ddots & ⋮\\ f\_1^{(n-1)}(x)& f\_2^{(n-1)}(x)& \cdots & f\_n^{(n-1)}(x)\end{array}\right|.$$

If the determinant’s result is $\ne 0$ at least at one point, the functions in the set are linearly independent from each other. Otherwise, if $W(f\_1,...,f\_n)=0$ for all $x$, the functions are linearly dependent.

Two Dimensional example

If testing a set of two equations $f\_1,f\_2$ the Wronskian is

$$W(f\_1,f\_2)=\left|\begin{array}{cc}f\_1(x)& f\_2(x)\\ f\_1^{\prime }(x)& f\_2^{\prime }(x)\end{array}\right|$$