# Wronskian

The Wronskian of a differential equation of the form y'' + p(t)y' + q(t)y = 0 is:

$W[y_1, y_2] = det \begin{bmatrix} y_1 & y_2 \\ y_1' & y_2' \end{bmatrix} = y_1 y_2' - y_2 y_1'$

where y1 and y2 are solutions of the said equation.

If the Wronskian is nonzero, it means y1 and y2 make up a fundamental set of solutions for the equation.

## Relation to Linear Algebra

Recall from linear algebra that, if the determinant of a matrix is nonzero, it means that the two columns of the matrix are linearly independent of each other. Thus, a nonzero Wronskian shows that the solutions y1 and y2 are linearly independent, or make up a fundamental set of solutions.

## Abel's Theorem

An alternate expression for the Wronskian (found by algebraic manipulation and similar processes):

$W[y_1, y_2] = ce^{-\int p(t)}$