Abel's identity

In mathematics, Abel's identity (also called Abel's formula 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 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 to 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 . In particular, when the differential equation is real-valued, the Wronskian is always either identically zero, always positive, or always negative at every point in (see proof below). The latter cases imply the two solutions and are linearly independent (see Wronskian for a proof). It is not necessary to assume that the second derivatives of the solutions and are continuous. Abel's theorem is particularly useful if , because it implies that is constant.