Wronskian
From Conservapedia
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'](images/math/f/4/e/f4e507ce31d7345fedc17148502280c2.png)
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)}](images/math/4/2/7/4270d6dca7cdb1be14a37bcfe8fcf189.png)
