In mathematics, Abel's identity (also called Abel's formula^[1] or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be generalised to nth-order linear ordinary differential equations. The identity is named after the Norwegian mathematician Niels Henrik Abel.

Since Abel's identity relates to the different linearly independent solutions of the differential equation, it can be used to find one solution from the other. It 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.

A generalisation of first-order systems of homogeneous linear differential equations is given by Liouville's formula.

Consider a homogeneous linear second-order ordinary differential equation

on an interval I of the real line with real- or complex-valued continuous functions p and q. Abel's identity states that the Wronskian of two real- or complex-valued solutions and of this differential equation, that is the function defined by the determinant

satisfies the relation

for each point .

• When the differential equation is real-valued, since is strictly positive, the Wronskian is always either identically zero, always positive, or always negative at every point in .
• If the two solutions and are linearly dependent, then the Wronskian is identically zero. Conversely, if the Wronskian is not zero at any point on the interval, then they are linearly independent.
• It is not necessary to assume that the second derivatives of the solutions and are continuous.
• If then is constant.

The Wronskian of functions on an interval is the function defined by the determinant

Consider a homogeneous linear ordinary differential equation of order :

on an interval of the real line with a real- or complex-valued continuous function . Let by solutions of this nth order differential equation. Then the generalisation of Abel's identity states that this Wronskian satisfies the relation:

for each point .

For brevity, we write for and omit the argument . It suffices to show that the Wronskian solves the first-order linear differential equation

because the remaining part of the proof then coincides with the one for the case .

In the case we have and the differential equation for coincides with the one for . Therefore, assume in the following.

The derivative of the Wronskian is the derivative of the defining determinant. It follows from the Leibniz formula for determinants that this derivative can be calculated by differentiating every row separately, hence

However, note that every determinant from the expansion contains a pair of identical rows, except the last one. Since determinants with linearly dependent rows are equal to 0, one is only left with the last one:

Since every solves the ordinary differential equation, we have

for every . Hence, adding to the last row of the above determinant times its first row, times its second row, and so on until times its next to last row, the value of the determinant for the derivative of is unchanged and we get

The solutions form the square-matrix valued solution

of the -dimensional first-order system of homogeneous linear differential equations

The trace of this matrix is , hence Abel's identity follows directly from Liouville's formula.

• 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. (1986). Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley.
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
• Weisstein, Eric W. "Abel's Differential Equation Identity". MathWorld.