Abel's Identity
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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 : y'' + p(x)y' + q(x)\,y = 0 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 W=(y_1,y_2) of two real- or complex-valued solutions y_1 and y_2 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 ...
: W(y_1,y_2)(x) =\beginy_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end =y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I, satisfies the relation : W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \cdot \exp\biggl(-\int_^x p(x') \,\textrmx'\biggr),\qquad x\in I, for every point ''x''0 in ''I''.


Remarks

* In particular, the Wronskian W(y_1,y_2) is either always the zero function or always different from zero with the same sign at every point x in I. In the latter case, the two solutions y_1 and y_2 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 y_1 and y_2 are continuous. * Abel's theorem is particularly useful if p(x)=0, because it implies that W 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 W for W(y_1,y_2) and omitting the argument x for brevity) : \begin W' &= y_1' y_2' + y_1 y_2'' - y_1'' y_2 - y_1' y_2' \\ & = y_1 y_2'' - y_1'' y_2. \end Solving for y'' in the original differential equation yields : y'' = -(py'+qy). Substituting this result into the derivative of the Wronskian function to replace the second derivatives of y_1 and y_2 gives : \begin W'&= -y_1(py_2'+qy_2)+(py_1'+qy_1)y_2 \\ &= -p(y_1y_2'-y_1'y_2)\\ &= -pW. \end 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 W(x_0) at x_0. Since the function p is continuous on I, it is bounded on every closed and bounded subinterval of I and therefore integrable, hence :V(x)=W(x) \exp\left(\int_^x p(\xi) \,\textrm\xi\right), \qquad x\in I, 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 :V'(x)=\bigl(W'(x)+W(x)p(x)\bigr)\exp\biggl(\int_^x p(\xi) \,\textrm\xi\biggr)=0,\qquad x\in I, due to the differential equation for W. Therefore, V has to be constant on I, 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 V(x_0) = W(x_0), Abel's identity follows by solving the definition of V for W(x).


Generalization

Consider a homogeneous linear nth-order (n \geq 1) ordinary differential equation : y^ + p_(x)\,y^ + \cdots + p_1(x)\,y' + p_0(x)\,y = 0, on an interval I of the real line with a real- or complex-valued continuous function p_. The generalisation of Abel's identity states that the Wronskian W(y_1,\ldots,y_n) of n real- or complex-valued solutions y_1,\ldots,y_n of this nth-order differential equation, that is the function defined by the determinant : W(y_1,\ldots,y_n)(x) =\begin y_1(x) & y_2(x) & \cdots & y_n(x)\\ y'_1(x) & y'_2(x)& \cdots & y'_n(x)\\ \vdots & \vdots & \ddots & \vdots\\ y_1^(x) & y_2^(x) & \cdots & y_n^(x) \end,\qquad x\in I, satisfies the relation : W(y_1,\ldots,y_n)(x)=W(y_1,\ldots,y_n)(x_0) \exp\biggl(-\int_^x p_(\xi) \,\textrm\xi\biggr),\qquad x\in I, for every point x_0 in I.


Direct proof

For brevity, we write W for W(y_1,\ldots,y_n) and omit the argument x. It suffices to show that the Wronskian solves the first-order linear differential equation :W'=-p_\,W, because the remaining part of the proof then coincides with the one for the case n=2. In the case n=1 we have W=y_1 and the differential equation for W coincides with the one for y_1. Therefore, assume n \geq 2 in the following. The derivative of the Wronskian W 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 : \beginW' & = \begin y'_1 & y'_2 & \cdots & y'_n\\ y'_1 & y'_2 & \cdots & y'_n\\ y''_1 & y''_2 & \cdots & y''_n\\ y_1 & y_2 & \cdots & y_n\\ \vdots & \vdots & \ddots & \vdots\\ y_1^ & y_2^ & \cdots & y_n^ \end + \begin y_1 & y_2 & \cdots & y_n\\ y''_1 & y''_2 & \cdots & y''_n\\ y''_1 & y''_2 & \cdots & y''_n\\ y_1 & y_2 & \cdots & y_n\\ \vdots & \vdots & \ddots & \vdots\\ y_1^ & y_2^ & \cdots & y_n^ \end\\ &\qquad+\ \cdots\ + \begin y_1 & y_2 & \cdots & y_n\\ y'_1 & y'_2 & \cdots & y'_n\\ \vdots & \vdots & \ddots & \vdots\\ y_1^ & y_2^ & \cdots & y_n^\\ y_1^ & y_2^ & \cdots & y_n^\\ y_1^ & y_2^ & \cdots & y_n^ \end.\end 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: : W'= \begin y_1 & y_2 & \cdots & y_n\\ y'_1 & y'_2 & \cdots & y'_n\\ \vdots & \vdots & \ddots & \vdots\\ y_1^ & y_2^ & \cdots & y_n^\\ y_1^ & y_2^ & \cdots & y_n^ \end. Since every y_i solves the ordinary differential equation, we have : y_i^ + p_\,y_i^ + \cdots + p_1\,y'_i + p_0\,y_i = -p_\,y_i^ for every i \in \lbrace 1,\ldots,n \rbrace. Hence, adding to the last row of the above determinant p_0 times its first row, p_1 times its second row, and so on until p_ times its next to last row, the value of the determinant for the derivative of W is unchanged and we get : W'= \begin y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ \vdots & \vdots & \ddots & \vdots \\ y_1^ & y_2^ & \cdots & y_n^ \\ -p_\,y_1^ & -p_\,y_2^ & \cdots & -p_\,y_n^ \end =-p_W.


Proof using Liouville's formula

The solutions y_1,\ldots,y_n form the square-matrix valued solution :\Phi(x)=\begin y_1(x) & y_2(x) & \cdots & y_n(x)\\ y'_1(x) & y'_2(x)& \cdots & y'_n(x)\\ \vdots & \vdots & \ddots & \vdots\\ y_1^(x) & y_2^(x) & \cdots & y_n^(x)\\ y_1^(x) & y_2^(x) & \cdots & y_n^(x) \end,\qquad x\in I, of the n-dimensional first-order system of homogeneous linear differential equations :\beginy'\\y''\\\vdots\\y^\\y^\end =\begin0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ -p_0(x)&-p_1(x)&-p_2(x)&\cdots&-p_(x)\end \beginy\\y'\\\vdots\\y^\\y^\end. The trace of this matrix is -p_(x), 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