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
of this differential equation, that is the function defined by the determinant satisfies the relation for each point
Differentiating the Wronskian using the product rule gives (writing
in the original differential equation yields Substituting this result into the derivative of the Wronskian function to replace the second derivatives of
gives This is a first-order linear differential equation, and it remains to show that Abel's identity gives the unique solution, which attains the value
Differentiating both sides, using the product rule, the chain rule, the derivative of the exponential function and the fundamental theorem of calculus, one obtains due to the differential equation for
, because otherwise we would obtain a contradiction to the mean value theorem (applied separately to the real and imaginary part in the complex-valued case).
, Abel's identity follows by solving the definition of
is either identically zero, always positive, or always negative, given that
is the function defined by the determinant Consider a homogeneous linear ordinary differential equation of order
of the real line with a real- or complex-valued continuous function
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
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
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 next to last row, the value of the determinant for the derivative of
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.