"Abel's formula" redirects here. For the formula on difference operators, see Summation by parts.
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 NorwegianmathematicianNiels Henrik Abel.
Since Abel's identity relates 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 to 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 intervalI of the real line with real- or complex-valued continuous functionsp 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 every point x0 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 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
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
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
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Abel s formula redirects here For the formula on difference operators see Summation by parts 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 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 to first order systems of homogeneous linear differential equations is given by Liouville s formula Contents 1 Statement 1 1 Remarks 1 2 Proof 2 Generalization 2 1 Direct proof 2 2 Proof using Liouville s formula 3 ReferencesStatement EditConsider a homogeneous linear second order ordinary differential equation y p x y q x y 0 displaystyle y p x y q x y 0 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 W y 1 y 2 displaystyle W y 1 y 2 of two real or complex valued solutions y 1 displaystyle y 1 and y 2 displaystyle y 2 of this differential equation that is the function defined by the determinant W y 1 y 2 x y 1 x y 2 x y 1 x y 2 x y 1 x y 2 x y 1 x y 2 x x I displaystyle W y 1 y 2 x begin vmatrix y 1 x amp y 2 x y 1 x amp y 2 x end vmatrix 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 exp x 0 x p x d x x I displaystyle W y 1 y 2 x W y 1 y 2 x 0 cdot exp biggl int x 0 x p x textrm d x biggr qquad x in I for every point x0 in I Remarks Edit In particular the Wronskian W y 1 y 2 displaystyle W y 1 y 2 is either always the zero function or always different from zero with the same sign at every point x displaystyle x in I displaystyle I In the latter case the two solutions y 1 displaystyle y 1 and y 2 displaystyle 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 displaystyle y 1 and y 2 displaystyle y 2 are continuous Abel s theorem is particularly useful if p x 0 displaystyle p x 0 because it implies that W displaystyle W is constant Proof Edit Differentiating the Wronskian using the product rule gives writing W displaystyle W for W y 1 y 2 displaystyle W y 1 y 2 and omitting the argument x displaystyle x for brevity 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 displaystyle begin aligned W amp y 1 y 2 y 1 y 2 y 1 y 2 y 1 y 2 amp y 1 y 2 y 1 y 2 end aligned Solving for y displaystyle y in the original differential equation yields y p y q y displaystyle y py qy Substituting this result into the derivative of the Wronskian function to replace the second derivatives of y 1 displaystyle y 1 and y 2 displaystyle y 2 gives W y 1 p y 2 q y 2 p y 1 q y 1 y 2 p y 1 y 2 y 1 y 2 p W displaystyle begin aligned W amp y 1 py 2 qy 2 py 1 qy 1 y 2 amp p y 1 y 2 y 1 y 2 amp pW end aligned 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 displaystyle W x 0 at x 0 displaystyle x 0 Since the function p displaystyle p is continuous on I displaystyle I it is bounded on every closed and bounded subinterval of I displaystyle I and therefore integrable hence V x W x exp x 0 x p 3 d 3 x I displaystyle V x W x exp left int x 0 x p xi textrm d xi right qquad x in I 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 V x W x W x p x exp x 0 x p 3 d 3 0 x I displaystyle V x bigl W x W x p x bigr exp biggl int x 0 x p xi textrm d xi biggr 0 qquad x in I due to the differential equation for W displaystyle W Therefore V displaystyle V has to be constant on I displaystyle I 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 V x 0 W x 0 displaystyle V x 0 W x 0 Abel s identity follows by solving the definition of V displaystyle V for W x displaystyle W x Generalization EditConsider a homogeneous linear n displaystyle n th order n 1 displaystyle n geq 1 ordinary differential equation y n p n 1 x y n 1 p 1 x y p 0 x y 0 displaystyle y n p n 1 x y n 1 cdots p 1 x y p 0 x y 0 on an interval I displaystyle I of the real line with a real or complex valued continuous function p n 1 displaystyle p n 1 The generalisation of Abel s identity states that the Wronskian W y 1 y n displaystyle W y 1 ldots y n of n displaystyle n real or complex valued solutions y 1 y n displaystyle y 1 ldots y n of this n displaystyle n th order differential equation that is the function defined by the determinant W y 1 y n x y 1 x y 2 x y n x y 1 x y 2 x y n x y 1 n 1 x y 2 n 1 x y n n 1 x x I displaystyle W y 1 ldots y n x begin vmatrix y 1 x amp y 2 x amp cdots amp y n x y 1 x amp y 2 x amp cdots amp y n x vdots amp vdots amp ddots amp vdots y 1 n 1 x amp y 2 n 1 x amp cdots amp y n n 1 x end vmatrix qquad x in I satisfies the relation W y 1 y n x W y 1 y n x 0 exp x 0 x p n 1 3 d 3 x I displaystyle W y 1 ldots y n x W y 1 ldots y n x 0 exp biggl int x 0 x p n 1 xi textrm d xi biggr qquad x in I for every point x 0 displaystyle x 0 in I displaystyle I Direct proof Edit For brevity we write W displaystyle W for W y 1 y n displaystyle W y 1 ldots y n and omit the argument x displaystyle x It suffices to show that the Wronskian solves the first order linear differential equation W p n 1 W displaystyle W p n 1 W because the remaining part of the proof then coincides with the one for the case n 2 displaystyle n 2 In the case n 1 displaystyle n 1 we have W y 1 displaystyle W y 1 and the differential equation for W displaystyle W coincides with the one for y 1 displaystyle y 1 Therefore assume n 2 displaystyle n geq 2 in the following The derivative of the Wronskian W displaystyle 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 W y 1 y 2 y n y 1 y 2 y n y 1 y 2 y n y 1 y 2 y n y 1 n 1 y 2 n 1 y n n 1 y 1 y 2 y n y 1 y 2 y n y 1 y 2 y n y 1 y 2 y n y 1 n 1 y 2 n 1 y n n 1 y 1 y 2 y n y 1 y 2 y n y 1 n 3 y 2 n 3 y n n 3 y 1 n 2 y 2 n 2 y n n 2 y 1 n y 2 n y n n displaystyle begin aligned W amp begin vmatrix y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n vdots amp vdots amp ddots amp vdots y 1 n 1 amp y 2 n 1 amp cdots amp y n n 1 end vmatrix begin vmatrix y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n vdots amp vdots amp ddots amp vdots y 1 n 1 amp y 2 n 1 amp cdots amp y n n 1 end vmatrix amp qquad cdots begin vmatrix y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n vdots amp vdots amp ddots amp vdots y 1 n 3 amp y 2 n 3 amp cdots amp y n n 3 y 1 n 2 amp y 2 n 2 amp cdots amp y n n 2 y 1 n amp y 2 n amp cdots amp y n n end vmatrix end aligned 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 y 1 y 2 y n y 1 y 2 y n y 1 n 2 y 2 n 2 y n n 2 y 1 n y 2 n y n n displaystyle W begin vmatrix y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n vdots amp vdots amp ddots amp vdots y 1 n 2 amp y 2 n 2 amp cdots amp y n n 2 y 1 n amp y 2 n amp cdots amp y n n end vmatrix Since every y i displaystyle y i solves the ordinary differential equation we have y i n p n 2 y i n 2 p 1 y i p 0 y i p n 1 y i n 1 displaystyle y i n p n 2 y i n 2 cdots p 1 y i p 0 y i p n 1 y i n 1 for every i 1 n displaystyle i in lbrace 1 ldots n rbrace Hence adding to the last row of the above determinant p 0 displaystyle p 0 times its first row p 1 displaystyle p 1 times its second row and so on until p n 2 displaystyle p n 2 times its next to last row the value of the determinant for the derivative of W displaystyle W is unchanged and we get W y 1 y 2 y n y 1 y 2 y n y 1 n 2 y 2 n 2 y n n 2 p n 1 y 1 n 1 p n 1 y 2 n 1 p n 1 y n n 1 p n 1 W displaystyle W begin vmatrix y 1 amp y 2 amp cdots amp y n y 1 amp y 2 amp cdots amp y n vdots amp vdots amp ddots amp vdots y 1 n 2 amp y 2 n 2 amp cdots amp y n n 2 p n 1 y 1 n 1 amp p n 1 y 2 n 1 amp cdots amp p n 1 y n n 1 end vmatrix p n 1 W Proof using Liouville s formula Edit The solutions y 1 y n displaystyle y 1 ldots y n form the square matrix valued solution F x y 1 x y 2 x y n x y 1 x y 2 x y n x y 1 n 2 x y 2 n 2 x y n n 2 x y 1 n 1 x y 2 n 1 x y n n 1 x x I displaystyle Phi x begin pmatrix y 1 x amp y 2 x amp cdots amp y n x y 1 x amp y 2 x amp cdots amp y n x vdots amp vdots amp ddots amp vdots y 1 n 2 x amp y 2 n 2 x amp cdots amp y n n 2 x y 1 n 1 x amp y 2 n 1 x amp cdots amp y n n 1 x end pmatrix qquad x in I of the n displaystyle n dimensional first order system of homogeneous linear differential equations y y y n 1 y n 0 1 0 0 0 0 1 0 0 0 0 1 p 0 x p 1 x p 2 x p n 1 x y y y n 2 y n 1 displaystyle begin pmatrix y y vdots y n 1 y n end pmatrix begin pmatrix 0 amp 1 amp 0 amp cdots amp 0 0 amp 0 amp 1 amp cdots amp 0 vdots amp vdots amp vdots amp ddots amp vdots 0 amp 0 amp 0 amp cdots amp 1 p 0 x amp p 1 x amp p 2 x amp cdots amp p n 1 x end pmatrix begin pmatrix y y vdots y n 2 y n 1 end pmatrix The trace of this matrix is p n 1 x displaystyle p n 1 x hence Abel s identity follows directly from Liouville s formula References Edit Rainville Earl David Bedient Phillip Edward 1969 Elementary Differential Equations Collier Macmillan International Editions Abel N H Precis d une theorie 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 Teschl Gerald 2012 Ordinary Differential Equations and Dynamical Systems Providence American Mathematical Society ISBN 978 0 8218 8328 0 Weisstein Eric W Abel s Differential Equation Identity MathWorld Retrieved from https en wikipedia org w index php title Abel 27s identity amp oldid 1092152351, wikipedia, wiki, book, books, library,