In mathematics, Abel's identity (also called Abel's differential equation identity) is an equation that expresses the Wronskian of two homogeneous solutions of a second-order linear ordinary differential equation in terms of the coefficients of the original differential equation. The identity is named after mathematician Niels Henrik Abel. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Nedstrand, near Finnøy where his father acted as rector. ...

Abel's identity, since it relates the different linearly independent solutions of the differential equation, can be used to find one solution from the other, provides useful identities relating the solutions, and is also useful as a part of other techniques such as the method of variation of parameters. It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly. In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ... In mathematics, variation of parameters or variation of constants is a method used to solve inhomogeneous linear ordinary differential equations. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to...

Definition

Given a homogeneous linear second-order ordinary differential equation A homogeneous differential equation is a first order ordinary differential equation with the form dy/dx = F(y/x). ...

Abel's identity can be written as

where W(x) is the Wronskian of the two linearly independent solutions to the differential equation. In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ...

Derivation

Let y₁ and y₂ be the two linearly independent solutions to the differential equation

Then the Wronskian of the two functions is defined as

Differentiating gives Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ...

Solving for y'' in the original differential equation yields

and the result is substituted into the Wronskian function:

This is a first-order linear differential equation.

References

• Abel, N. H., "Précis d'une théorie des fonctions elliptiques" J. Reine Angew. Math. , 4 (1829) pp. 309–348.
• Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, 1986.
• Weisstein, Eric W., "Abel's Differential Equation Identity", From MathWorld--A Wolfram Web Resource.