Wronskian

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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)}

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