In
mathematics, Abel's identity (also called Abel's formula
or Abel's differential equation identity) is an equation that expresses the
Wronskian
In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.
Definition
The Wronskian o ...
of two solutions of a homogeneous second-order linear
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
in terms of a coefficient of the original differential equation.
The relation can be generalised to ''n''th-order linear ordinary differential equations. The identity is named after the
Norwegian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
.
Since Abel's identity relates the different
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
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
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible ...
. It is especially useful for equations such as
Bessel's equation
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
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
In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the ...
.
Statement
Consider a
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
linear second-order ordinary differential equation
:
on an
interval ''I'' of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (201 ...
- or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
:
satisfies the relation
:
for every point ''x''
0 in ''I''.
Remarks
* In particular, the Wronskian
is either always the zero function or always different from zero with the same sign at every point
in
. In the latter case, the two solutions
and
are linearly independent (see the article about the 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
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
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
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
, the derivative of the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
and the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
, 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
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It ...
(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
Consider a homogeneous linear
th-order (
) ordinary differential equation
:
on an interval
of the real line with a real- or complex-valued continuous function
. The generalisation of Abel's identity states that the Wronskian
of
real- or complex-valued solutions
of this
th-order differential equation, that is the function defined by the determinant
:
satisfies the relation
:
for every point
in
.
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
In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the ...
.
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.
*
* {{MathWorld, urlname=AbelsDifferentialEquationIdentity, title=Abel's Differential Equation Identity
Articles containing proofs
Mathematical identities
Ordinary differential equations
Niels Henrik Abel