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:
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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.
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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 ==

Latest revision as of 18:21, 24 April 2013

The Wronskian of a differential equation of the form is:

where and are solutions of the said equation.

If the Wronskian is nonzero, it means and 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 and 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):