Difference between revisions of "Wronskian"
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| − | The '''Wronskian''' of a differential equation of the form <math>y'' + p(t)y' + q(t)y = 0</math> is: | + | 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] = | <math>W[y_1, y_2] = | ||
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== Relation to Linear Algebra == | == 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. | + | 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 == | == Abel's Theorem == | ||
Revision as of 22:21, April 24, 2013
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>