Abel's identity

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 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.
Statement
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 .
Remarks
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.
Proof
Differentiating the Wronskian using the product rule gives (writing for and omitting the argument for brevity)
Solving for in the original differential equation yields
Substituting this result into the derivative of the Wronskian function to replace the second derivatives of and 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 at . Since the function is continuous on , it is bounded on every closed and bounded subinterval of and therefore integrable, hence
is a well-defined function. 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 . Therefore, has to be constant on , 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). Since , Abel's identity follows by solving the definition of for .
Generalization
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 .
Direct proof
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
Proof using Liouville's formula
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.
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. (1986). Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley.
Articles containing proofs
Mathematical identities
Ordinary differential equations
Niels Henrik Abel