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Jacobi Differential Equation

(1-x^2)y^('')+[beta-alpha-(alpha+beta+2)x]y^'+n(n+alpha+beta+1)y=0
(1)

or

d/(dx)[(1-x)^(alpha+1)(1+x)^(beta+1)y^']+n(n+alpha+beta+1)(1-x)^alpha(1+x)^betay=0.
(2)

The solutions are Jacobi polynomials P_n^((alpha,beta))(x) or, in terms of hypergeometric functions, as

y(x)=C_1_2F_1(-n,n+1+alpha+beta,1+alpha,1/2(x-1)) +2^alpha(x-1)^(-alpha)C_2_2F_1(-n-alpha,n+1+beta,1-alpha,1/2(1-x)).
(3)

The equation (2) can be transformed to

(d^2u)/(dx^2)+[1/4(1-alpha^2)/((1-x)^2)+1/4(1-beta^2)/((1+x)^2)+(n(n+alpha+beta+1)+1/2(alpha+1)(beta+1))/(1-x^2)]u=0,
(4)

where

u(x)=(1-x)^((alpha+1)/2)(1+x)^((beta+1)/2)P_n^((alpha,beta))(x),
(5)

and

(d^2u)/(dtheta^2)+[(1/4-alpha^2)/(4sin^2(1/2theta))+(1/4-beta^2)/(4cos^2(1/2theta))+(n+(alpha+beta+1)/2)^2]u=0,
(6)

where

u(theta)=sin^(alpha+1/2)(1/2theta)cos^(beta+1/2)(1/2theta)P_n^((alpha,beta))(costheta).
(7)

Zwillinger (1997, p. 123) gives a related differential equation he terms Jacobi's equation

x(1-x)y^('')+[gamma-(alpha+1)x]y^'+n(alpha+n)y=0
(8)

(Iyanaga and Kawada 1980, p. 1480), which has solution

y=C_1_2F_1(-n,n+alpha,gamma,x) -(-1)^(-gamma)x^(1-gamma)C_2_2F_1(1-n-gamma,1+n+alpha-gamma,2-gamma,x).
(9)

Zwillinger (1997, p. 120; duplicated twice) also gives another types of ordinary differential equation called a Jacobi equation,

(a_1+b_1x+c_1y)(xy^'-y)-(a_2+b_2x+c_2y)y^' +(a_3+b_3x+c_3y)=0
(10)

(Ince 1956, p. 22).

In the calculus of variations, the partial differential equation

d/(dx)Omega_(eta^')-Omega_eta=d/(dx)(f_(y^'y)eta+f_(y^'y)eta^')-(f_(yy)eta+f_(yy^')eta^')=0,
(11)

where

Omega(x,eta,eta^')=1/2(f_(yy)eta^2+2f_(yy^')etaeta^'+f_(y^'y)eta^('2))
(12)

is called the Jacobi differential equation.

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References

Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 162-163, 1925.Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 22, 1956.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.

Referenced on Wolfram|Alpha

Jacobi Differential Equation

Cite this as:

Weisstein, Eric W. "Jacobi Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiDifferentialEquation.html

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