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

Suppose and are two solutions of a 2nd order differential equation, in the form mentioned above.

is always zero or never zero (in the intended range of solutions).

Note

If the Wronskian is non-zero at a point and zero on another point, cannot be solutions to the same differential equation.

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 .

Abel’s forumla

The conclusion in the above proof is known as the Abel’s formula.

Theorem 2

Note

This theorem is valid only under some conditions, which are out of the scope of the course and not covered.