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Chebyshev Differential Equation

(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+alpha^2y=0
(1)

for |x|<1. The Chebyshev differential equation has regular singular points at -1, 1, and infty. It can be solved by series solution using the expansions

y=sum_(n=0)^(infty)a_nx^n
(2)
y^'=sum_(n=0)^(infty)na_nx^(n-1)
(3)
=sum_(n=1)^(infty)na_nx^(n-1)
(4)
=sum_(n=0)^(infty)(n+1)a_(n+1)x^n
(5)
y^('')=sum_(n=0)^(infty)(n+1)na_(n+1)x^(n-1)
(6)
=sum_(n=1)^(infty)(n+1)na_(n+1)x^(n-1)
(7)
=sum_(n=0)^(infty)(n+2)(n+1)a_(n+2)x^n.
(8)

Now, plug equations (6) and (8) into the original equation (◇) to obtain

(1-x^2)sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n -xsum_(n=0)^infty(n+1)n_(n+1)x^n+alpha^2sum_(n=0)^inftya_nx^n=0
(9)
sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^(n+2) -sum_(n=0)^infty(n+1)a_(n+1)x^(n+1)+alpha^2sum_(n=0)^inftya_nx^n=0
(10)
sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=2)^inftyn(n-1)a_nx^n -sum_(n=1)^inftyna_nx^n+alpha^2sum_(n=0)^inftya_nx^n=0
(11)
2·1a_2+3·2a_3x-1·ax+alpha^2a_0+alpha^2a_1x +sum_(n=2)^infty[(n+2)(n+1)a_(n+2)-n(n-1)a_n-na_n+alpha^2a_n]x^n=0
(12)
(2a_2+alpha^2a_0)+[(alpha^2-1)a_1+6a_3]x +sum_(n=2)^infty[(n+2)(n+1)a_(n+2)+(alpha^2-n^2)a_n]x^n=0,
(13)

so

2a_2+alpha^2a_0=0
(14)
(alpha^2-1)a_1+6a_3=0,
(15)

and by induction,

a_(n+2)=(n^2-alpha^2)/((n+1)(n+2))a_n
(16)

for n=2, 3, ....

Since (14) and (15) are special cases of (16), the general recurrence relation can be written

a_(n+2)=(n^2-alpha^2)/((n+1)(n+2))a_n
(17)

for n=0, 1, .... From this, we obtain for the even coefficients

a_2=(-alpha^2)/2a_0
(18)
a_4=(2^2-alpha^2)/(3·4)a_2=((2^2-alpha^2)(-alpha^2))/(1·2·3·4)a_0
(19)
a_(2n)=([(2n)^2-alpha^2][(2n-2)^2-alpha^2]...(-alpha^2))/((2n)!)a_0,
(20)

and for the odd coefficients

a_3=(1-alpha^2)/6a_1
(21)
a_5=(3^2-alpha^2)/(4·5)a_3=((3^2-alpha^2)(1^2-alpha^2))/(5!)a_1
(22)
a_(2n-1)=([(2n-1)^2-alpha^2][(2n-3)^2-alpha^2]...[1^2-alpha^2])/((2n+1)!)a_1.
(23)

The even coefficients k=2n can be given in closed form as

a_(k even)=a_0product_(j=1)^(k/2)(k-2j)^2-alpha^2
(24)
=(2^(k-1)pialphacsc(1/2pialpha))/(Gamma(1-1/2k-1/2alpha)Gamma(1-1/2k+1/2alpha))a_0,
(25)

and the odd coefficients k=2n-1 as

a_(k odd)=a_1product_(j=1)^((k-1)/2)(k-2j)^2-alpha^2
(26)
=(2^(k-1)pialphasec(1/2pialpha))/(Gamma(1-1/2k-1/2alpha)Gamma(1-1/2k+1/2alpha))a_1.
(27)

The general solution is then given by summing over all indices,

y=a_0[1+sum_(k=2,4,...)^infty(a_(k even))/(k!)x^k] +[x+sum_(k=3,5,...)^infty(a_(k odd))/(k!)x^k],
(28)

which can be done in closed form as

y=a_0cos(alphasin^(-1)x)+(a_1)/alphasin(alphasin^(-1)x).
(29)

Performing a change of variables gives the equivalent form of the solution

y=b_1cos(alphacos^(-1)x)+b_2sin(alphacos^(-1)x)
(30)
=b_1T_alpha(x)+b_2sqrt(1-x^2)U_(alpha-1)(x),
(31)

where T_n(x) is a Chebyshev polynomial of the first kind and U_n(x) is a Chebyshev polynomial of the second kind. Another equivalent form of the solution is given by

y=c_1cosh[alphaln(x+sqrt(x^2-1))] +ic_2sinh[alphaln(x+sqrt(x^2-1))].
(32)

See also

Chebyshev Polynomial of the First Kind, Chebyshev Polynomial of the Second Kind

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 735, 1985.Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 232 and 252, 1986.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997.

Referenced on Wolfram|Alpha

Chebyshev Differential Equation

Cite this as:

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

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