The Wronskian of a System

The Wronskian () of the system is the determinant of matrix defined below:

where and are components of the general solution to our current differential equation:

where and are constants.

As described above the Wronskian of the system is the determinant of this matrix:

Sources and more information:

Paul's Online Notes - More on the Wronskian
1-Second Order Scalar DE (Sect 3-1 to 3-5)
Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 118

Abel's Formula

If the Wronskian of the system at any does not equal 0 then the Wronskian of the system at does not equal zero, therefore the matrix is a fundamental matrix for the system:

Sources and more information:

Fundamental Set of Solutions, Fundamental Matrix and Abel's Formula (sect 4.3 and 4.4)
1-Second Order Scalar DE (Sect 3-1 to 3-5)

For Second Order Linear Systems

Given a second order linear system differential equations expand, diffeq 13-1