Wronskian

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

<math>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'</math>

where <math>y_1</math> and <math>y_2</math> are solutions of the said equation.

If the Wronskian is nonzero, it means <math>y_1</math> and <math>y_2</math> 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 <math>y_1</math> and <math>y_2</math> 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):

<math>W[y_1, y_2] = ce^{-\int p(t)}</math>