Abel's identity

The abelian identity is an expression for the Wronskian of two linearly independent solutions of a linear homogeneous ordinary differential equation of second order. The relationship was derived in 1827 by the Norwegian mathematician Niels Henrik Abel.

Statement

Consider the linear ordinary differential equation of second order

Then for the Wronskian of two solutions of the differential equation

Evidence

By definition, in which a fundamental system of the differential equation

Is. According to the Liouville formula applies

Application

The abelian identity allows to calculate the Wronskian for a known value at the position for all others. In particular, the Wronskian is constant, if valid. Because of the relationship that the Wronskian establishes between two linearly independent solutions, it allows an attacker to calculate the one from the other.