Abelian identity

This article deals with the Abelian identity for the Wronsky determinant. For the Abelian identity as a transformation of a sum of products of two factors, see abelian partial summation .

The Abelian identity is an expression for the Wronsky determinant of two linearly independent homogeneous solutions of a linear ordinary differential equation of the second order. The relationship was derived in 1827 by the Norwegian mathematician Niels Henrik Abel (1802-1829).

statement

Let the linear ordinary differential equation of the second order be given

${\displaystyle y''(x)-a_{1}(x)y'(x)-a_{0}(x)y(x)=0}$.

For the Wronsky determinant of two solutions of the differential equation then applies

${\displaystyle W(x)=W(x_{0})\ \exp \left(\int _{x_{0}}^{x}a_{1}(\xi )\,\mathrm {d} \xi \right)}$.

proof

By definition , where is a fundamental system for the differential equation ${\displaystyle W(x)=\det \Phi (x)}$ ${\displaystyle \Phi }$

${\displaystyle Y'(x)=A(x)Y(x)}$ With ${\displaystyle A(x):={\begin{pmatrix}0&1\\a_{0}(x)&a_{1}(x)\\\end{pmatrix}}}$

is. According to Liouville's formula ,

${\displaystyle W(x)=W(x_{0})\ \exp \left(\int _{x_{0}}^{x}\mathrm {Spur} (A(\xi ))\ \mathrm {d} \xi \right)=W(x_{0})\ \exp \left(\int _{x_{0}}^{x}a_{1}(\xi )\,\mathrm {d} \xi \right)}$.

${\displaystyle \Box }$

application

The Abelian identity allows the Wronsky determinant to be calculated for all others if the value is known at that point . In particular, the Wronsky determinant is constant if holds. Due to the relationship that the Wronsky determinant establishes between two linearly independent solutions, it may allow one to be calculated from the other. ${\displaystyle x_{0}}$ ${\displaystyle x}$ ${\displaystyle a_{1}(x)\equiv 0}$

literature

• W. Boyce, R. Di Prima: Elementary differential equations and boundary value problems. Wiley, New York 1969.
• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).

Web links

• Eric W. Weisstein : Abel's Differential Equation Identity . In: MathWorld (English).