In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Abel's identity (also called Abel's differential equation identity) is an equation that expresses the Wronskian

Wronskian

In mathematics, the Wronskian is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes be used to show that a set of solutions is linearly independent.-Definition:...

 of two homogeneous solutions of a second-order linear ordinary differential equation

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

 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

Norway

Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Niels Henrik Abel

Niels Henrik Abel

Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

. Since Abel's identity relates the different linearly independent

Linear independence

In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...

 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

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...

. It is especially useful for equations such as Bessel's equation

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential 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

Liouville's formula

In mathematics, Liouville's formula 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 sum of the diagonal coefficients of the system...

.

Statement of Abel's identity

Consider a homogeneous

Homogeneous differential equation

The term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...

 linear second-order ordinary differential equation NEWLINE

NEWLINE

$y\; +\; p(x)y\text{'}\; +\; q(x)\backslash ,y\; =\; 0$

NEWLINE on an interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 I of the real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

 with a real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

- or complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

-valued continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 p. Abel's identity states that the Wronskian W(y₁,y₂) of two real- or complex-valued solutions y₁ and y₂ of this differential equation, that is the function defined by the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

NEWLINE

NEWLINE

$W(y\_1,y\_2)(x)y\_1(x)\backslash ,y\text{'}\_2(x)\; -\; y\text{'}\_1(x)\backslash ,y\_2(x),\backslash qquad\; x\backslash in\; I,$

NEWLINE satisfies the relation NEWLINE

NEWLINE

$W(y\_1,y\_2)(x)=W(y\_1,y\_2)(x\_0)\; \backslash exp\backslash biggl(-\backslash int\_\{x\_0\}^x\; p(\backslash xi)\; \backslash ,\backslash textrm\{d\}\backslash xi\backslash biggr),\backslash qquad\; x\backslash in\; I,$

NEWLINE for every point x₀ in I.

Remarks

NEWLINE

NEWLINE
• In particular, the Wronskian W(y₁,y₂) is either the zero function or it is different from zero at every point x in I. In the latter case, the two solutions y₁ and y₂ are linearly independent (see that article about the Wronskian for a proof).
NEWLINE
• It is not necessary to assume that the second derivatives of the solutions y₁ and y₂ are continuous.

NEWLINE

Proof of Abel's identity

Differentiating

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 the Wronskian using the product rule

Product rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

 gives (writing W for W(y₁,y₂) and omitting the argument x for brevity) NEWLINE

NEWLINE

$$

NEWLINE \begin{align} 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{align} Solving for $y$ in the original differential equation yields NEWLINE

NEWLINE

$y\; =\; -(py\text{'}+qy).\; \backslash ,$

NEWLINE Substituting this result into the derivative of the Wronskian function to replace the second derivatives of y₁ and y₂ givesNEWLINE

NEWLINE

$$

NEWLINE \begin{align} W'&= -y_1(py_2'+qy_2)+(py_1'+qy_1)y_2 \\ &= -p(y_1y_2'-y_1'y_2)\\ &= -pW. \end{align} 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(x₀) at x₀. Since the function p is continuous on I, it is bounded on every closed and bounded subinterval of I and therefore integrable, hence $V(x)=W(x)\; \backslash exp\backslash left(\backslash int\_\{x\_0\}^x\; p(\backslash xi)\; \backslash ,\backslash textrm\{d\}\backslash xi\backslash right),\; \backslash qquad\; x\backslash in\; I,$ is a well-defined function. Differentiating both sides, using the product rule, the chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

, the derivative of the exponential function

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 and the fundamental theorem of calculus

Fundamental theorem of calculus

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

, we obtain $V\text{'}(x)=\backslash bigl(W\text{'}(x)+W(x)p(x)\backslash bigr)\backslash exp\backslash biggl(\backslash int\_\{x\_0\}^x\; p(\backslash xi)\; \backslash ,\backslash textrm\{d\}\backslash xi\backslash biggr)=0,\backslash qquad\; x\backslash in\; I,$ 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

Mean value theorem

In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...

 (applied separately to the real and imaginary part in the complex-valued case). Since V(x₀) = W(x₀), Abel's identity follows by solving the definition of V for W(x).

Generalisation of Abel's identity

Consider a homogeneous linear n^th-order (n ≥ 1) ordinary differential equation NEWLINE

NEWLINE

$y^\{(n)\}\; +\; p\_\{n-1\}(x)\backslash ,y^\{(n-1)\}\; +\; \backslash cdots\; +\; p\_1(x)\backslash ,y\text{'}\; +\; p\_0(x)\backslash ,y\; =\; 0,$

NEWLINE on an interval I of the real line with a real- or complex-valued continuous function p_n−1. The generalisation of Abel's identity states that the Wronskian W(y₁,…,y_n) of n real- or complex-valued solutions y₁,…,y_n of this n^th-order differential equation, that is the function defined by the determinant NEWLINE

NEWLINE

$W(y\_1,\backslash ldots,y\_n)(x)\backslash begin\{vmatrix\}$

NEWLINE 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^{(n-1)}(x) & y_2^{(n-1)}(x) & \cdots & y_n^{(n-1)}(x) \end{vmatrix},\qquad x\in I, satisfies the relation NEWLINE

NEWLINE

$W(y\_1,\backslash ldots,y\_n)(x)=W(y\_1,\backslash ldots,y\_n)(x\_0)\; \backslash exp\backslash biggl(-\backslash int\_\{x\_0\}^x\; p\_\{n-1\}(\backslash xi)\; \backslash ,\backslash textrm\{d\}\backslash xi\backslash biggr),\backslash qquad\; x\backslash in\; I,$

NEWLINE for every point x₀ in I.

Direct proof

For brevity, we write W for W(y₁,…,y_n) and omit the argument x. It suffices to show that the Wronskian solves the first-order linear differential equation $W\text{'}=-p\_\{n-1\}\backslash ,W,$ 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 = y₁ and the differential equation for W coincides with the one for y₁. Therefore, assume n ≥ 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 NEWLINE

NEWLINE

NEWLINE \begin{vmatrix} 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^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)} \end{vmatrix} + \begin{vmatrix} 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^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)} \end{vmatrix}\\ &\qquad+\ \cdots\ + \begin{vmatrix} y_1 & y_2 & \cdots & y_n\\ y'_1 & y'_2 & \cdots & y'_n\\ \vdots & \vdots & \ddots & \vdots\\ y_1^{(n-3)} & y_2^{(n-3)} & \cdots & y_n^{(n-3)}\\ y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)}\\ y_1^{(n)} & y_2^{(n)} & \cdots & y_n^{(n)} \end{vmatrix}.\end{align} 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, we're only left with the last one: NEWLINE

NEWLINE

$W\text{'}=$

NEWLINE \begin{vmatrix} y_1 & y_2 & \cdots & y_n\\ y'_1 & y'_2 & \cdots & y'_n\\ \vdots & \vdots & \ddots & \vdots\\ y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)}\\ y_1^{(n)} & y_2^{(n)} & \cdots & y_n^{(n)} \end{vmatrix}. Since every y_i solves the ordinary differential equation, we have NEWLINE

NEWLINE

$y\_i^\{(n)\}\; +\; p\_\{n-2\}\backslash ,y\_i^\{(n-2)\}\; +\; \backslash cdots\; +\; p\_1\backslash ,y\text{'}\_i\; +\; p\_0\backslash ,y\_i\; =\; -p\_\{n-1\}\backslash ,y\_i^\{(n-1)\}$

NEWLINE for every i ∈ {1,...,n}. Hence, adding to the last row of the above determinant p₀ times its first row, p₁ times its second row, and so on until p_n−2 times its next to last row, the value of the determinant for the derivative of W is unchanged and we get NEWLINE

NEWLINE

$W\text{'}=$

NEWLINE \begin{vmatrix} y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)} \\ -p_{n-1}\,y_1^{(n-1)} & -p_{n-1}\,y_2^{(n-1)} & \cdots & -p_{n-1}\,y_n^{(n-1)} \end{vmatrix}

Proof using Liouville's formula

The solutions y₁,…,y_n form the square-matrix valued solution of the n-dimensional first-order system of homogeneous linear differential equations The trace

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

of this matrix is −p_n−1(x), hence Abel's identity follows directly from Liouville's formula.