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 $y_{1}$ and $y_{2}$ are solutions of the said equation.

If the Wronskian is nonzero, it means $y_{1}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2} 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):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W[y_1, y_2] = ce^(-\int p(t))}