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 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

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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

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  • 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

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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  .

Proof that the Wronskian never changes sign

For all  , the Wronskian   is either identically zero, always positive, or always negative, given that  ,  , and   are real-valued. This is demonstrated as follows.

Abel's identity states that  

Let  . Then   must be a real-valued constant because   and   are real-valued.

Let  . As   is real-valued, so is  , so   is strictly positive.

Thus,   is identically zero when  , always positive when   is positive, and always negative when   is negative.

Furthermore, when  ,  , and  , one can similarly show that   is either identically   or non-zero for all values of x.

Generalization

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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

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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

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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

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  1. ^ Rainville, Earl David; Bedient, Phillip Edward (1969). Elementary Differential Equations. Collier-Macmillan International Editions.