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Created page with '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_…'
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>
<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>