Wronskian

Consider the equation, where are functions of alone, and which has fundamental solutions :

The Wronskian of two solutions of differential equation, is defined to be:

Theorem 1

The Wronskian of two solutions of the above differential equation is identically zero or never zero.

Note

Identically zero means the function is always zero.

Proof

Consider the equation, where are functions of alone.

Let be fundamental solutions of the equation:

By solving the above relation:

Suppose there exists such that . That implies . That implies is always .

Theorem 2

The solutions of the above differential equation are linearly dependent iff their Wronskian vanish identically.