Abel's identity

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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 differential equation identity) is an equation that expresses the Wronskian of two homogeneous solutions of a second-order linear ordinary differential equation in terms of the coefficients of the original differential equation. The identity is named after mathematician Niels Henrik Abel.

Abel's identity, since it relates the different linearly independent solutions of the differential equation, can be used to find one solution from the other, 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.

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[edit] Definition

Consider a homogeneous linear second-order ordinary differential equation

\frac{\textrm{d}^2y}{\textrm{d}x^2} + P(x)\frac{\textrm{d}y}{\textrm{d}x} + Q(x)\,y = 0

on an interval I of the real line with a continuous function P. Abel's identity states that the Wronskian W of two solutions of the differential equation satisfies the relation

W(x)=W(x_0) \exp\left(-\int_{x_0}^x P(\xi) \,\textrm{d}\xi\right),\qquad x\in I,

for every point x0 in I.

In particular, the Wronskian is either the zero function or it is different from zero at every point x in I. In the latter case, the two solutions are linearly independent.

[edit] Derivation

Let y1 and y2 denote two solutions to the differential equation

y'' + P(x)\,y' + Q(x)\,y = 0.

Then the Wronskian of the two functions is defined as

W(x) = y_1(x) y_2'(x) - y_1'(x) y_2(x),\qquad x\in I.

Differentiating using the product rule gives (omitting the argument x for brevity)

\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

y'' = -(Py'+Qy). \,

Substituting this result into the derivative of the Wronskian function:

\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. Define

V(x)=W(x) \exp\left(\int_{x_0}^x P(\xi) \,\textrm{d}\xi\right), \qquad x\in I.

Differentiating both sides, using the product rule, the chain rule, the derivative of the exponential function and the fundamental theorem of calculus, we obtain

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. Therefore, V has to be constant on I, because otherwise we would obtain a contradiction to the mean value theorem. Since V(x0) = W(x0), Abel's identity follows by solving the definition of V for W(x).

[edit] Generalisation

For an nth order equation of the form

y^{(n)} + p_1(x)\,y^{(n-1)}+ \cdots + p_{n-1}(x)\,y = 0,

the Wronskian is given by

W(x)=W(x_0) \exp\left(-\int_{x_0}^x p_1(\xi) \,\textrm{d}\xi\right)

[edit] 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. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, 1986.
  • Weisstein, Eric W., "Abel's Differential Equation Identity", From MathWorld--A Wolfram Web Resource.