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Kelvin Differential Equation

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The complex second-order ordinary differential equation

x^2y^('')+xy^'-(ix^2+nu^2)y=0
(1)

(Abramowitz and Stegun 1972, p. 379; Zwillinger 1997, p. 123), whose solutions can be given in terms of the Kelvin functions as

y=ber_nu(x)+ibei_nu(x)
(2)
=ber_(-nu)(x)+ibei_(-nu)(x)
(3)
=ker_nu(x)+ikei_nu(x)
(4)
=ker_(-nu)(x)+ikei_(-nu)(x)
(5)

(Abramowitz and Stegun 1972, p. 379).

The general solution is

y(x)=c_1J_nu(-(-1)^(3/4)z)+c_2Y_nu(-(-1)^(3/4)z),
(6)

where J_nu(z) is a Bessel function of the first kind and Y_nu(z) is a Bessel function of the second kind.

See also

Kelvin Functions

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997.

Referenced on Wolfram|Alpha

Kelvin Differential Equation

Cite this as:

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

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