Abel's identity

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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.

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.

[edit] Definition

Given a homogeneous linear second-order ordinary differential equation

$\frac{\textrm{d}^2y}{\textrm{d}x^2} + P(x)\frac{\textrm{d}y}{\textrm{d}x} + Q(x)\,y = 0.$

Abel's identity can be written as

$W(x)=W(0) \exp\left(-\int_0^x P(\xi) \,\textrm{d}\xi\right)$

where W(x) is the Wronskian of the two linearly independent solutions to the differential equation.

[edit] Derivation

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

$y'' + P(x)\,y' + Q(x)\,y = 0,$

Then the Wronskian of the two functions is defined as

$W(x) = y_1' y_2 - y_1 y_2'. \,$

Differentiating gives

$W'(x) = y_1'' y_2 + y_1' y_2' - y_1' y_2' - y_2'' y_1 \,$

$W'(x) = y_1'' y_2 - y_1 y_2''. \,$

Solving for $y''$ in the original differential equation yields

$y'' = -P(x)\,y'-Q(x)\,y. \,$

and the result is substituted into the Wronskian function:

$W'(x) = \left(-P(x)y_1'-Q(x)\,y_1\right)y_2-y_1\left(-P(x)y_2'-Q(x)\,y_2\right) \,$

$W'(x) = -P(x)y_1'y_2 - Q(x)y_1y_2 + P(x)y_1y_2' + Q(x)y_1y_2 \,$

$W'(x) = -P(x)(y_1'y_2-y_1y_2') \,$

$W'(x) = -P(x) \, W(x)$

This is a first-order linear differential equation.

$\frac{\textrm{d}W}{W} = -P(x)\,\textrm{d}x \,$

$\ln\left(\frac{W(x)}{W(0)}\right)=-\int_0^x P(\xi)\,\textrm{d}\xi \,$

$W(x)=W(0) \exp\left(-\int_0^x P(\xi) \,\textrm{d}\xi\right).$

[edit] 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.

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Categories: Mathematical identities | Ordinary differential equations